# differentiability implies continuity

If you have trouble accessing this page and need to request an alternate format, contact ximera@math.osu.edu. Differentiability Implies Continuity We'll show that if a function is differentiable, then it's continuous. Theorem 1.1 If a function f is differentiable at a point x = a, then f is continuous at x = a. Practice: Differentiability at a point: graphical, Differentiability at a point: algebraic (function is differentiable), Differentiability at a point: algebraic (function isn't differentiable), Practice: Differentiability at a point: algebraic, Proof: Differentiability implies continuity. © 2013–2020, The Ohio State University — Ximera team, 100 Math Tower, 231 West 18th Avenue, Columbus OH, 43210–1174. Well a lack of continuity would imply one of two possibilities: 1: The limit of the function near x does not exist. If is differentiable at , then is continuous at . looks like a “vertical tangent line”, or if it rapidly oscillates near a, then the function Differentiability and continuity : If the function is continuous at a particular point then it is differentiable at any point at x=c in its domain. Thus, Therefore, since is defined and , we conclude that is continuous at . The best thing about differentiability is that the sum, difference, product and quotient of any two differentiable functions is always differentiable. Recall that the limit of a product is the product of the two limits, if they both exist. Differential coefficient of a function y= f(x) is written as d/dx[f(x)] or f' (x) or f (1)(x) and is defined by f'(x)= limh→0(f(x+h)-f(x))/h f'(x) represents nothing but ratio by which f(x) changes for small change in x and can be understood as f'(x) = lim?x→0(? whenever the denominator is not equal to 0 (the quotient rule). Fractals , for instance, are quite “rugged” $($see first sentence of the third paragraph: “As mathematical equations, fractals are … Nevertheless, Darboux's theorem implies that the derivative of any function satisfies the conclusion of the intermediate value theorem. This theorem is often written as its contrapositive: If f(x) is not continuous at x=a, then f(x) is not differentiable at x=a. We want to show that is continuous at by showing that . If $f$ is differentiable at $a,$ then it is continuous at $a.$ Proof Suppose that $f$ is differentiable at the point $x = a.$ Then we know that However, continuity and … continuous on \RR . In such a case, we If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Proof that differentiability implies continuity. Differentiability Implies Continuity: SHARP CORNER, CUSP, or VERTICAL TANGENT LINE So, differentiability implies continuity. exist, for a different reason. Continuity And Differentiability. Browse more videos. If you're seeing this message, it means we're having trouble loading external resources on our website. There are connections between continuity and differentiability. In figure B \lim _{x\to a^{+}} \frac {f(x)-f(a)}{x-a}\ne \lim _{x\to a^{-}} \frac {f(x)-f(a)}{x-a}. FALSE. Playing next. So now the equation that must be satisfied. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. UNIFORM CONTINUITY AND DIFFERENTIABILITY PRESENTED BY PROF. BHUPINDER KAUR ASSOCIATE PROFESSOR GCG-11, CHANDIGARH . Just remember: differentiability implies continuity. Continuously differentiable functions are sometimes said to be of class C 1. Proof: Differentiability implies continuity. Then. we must show that \lim _{x\to a} f(x) = f(a). Derivatives from first principle The Infinite Looper. Hence, a function that is differentiable at $$x = a$$ will, up close, look more and more like its tangent line at $$( a , f ( a ) )$$, and thus we say that a function is differentiable at $$x = a$$ is locally linear. Starting with \lim _{x\to a} \left (f(x) - f(a)\right ) we multiply and divide by (x-a) to get. Class 12 Maths continuity and differentiability Exercise 5.1 to Exercise 5.8, and Miscellaneous Questions NCERT Solutions are extremely helpful while doing your homework or while preparing for the exam. Here is a famous example: 1In class, we discussed how to get this from the rst equality. It follows that f is not differentiable at x = 0.. Finding second order derivatives (double differentiation) - Normal and Implicit form. • If f is differentiable on an interval I then the function f is continuous on I. Sal shows that if a function is differentiable at a point, it is also continuous at that point. Get Free NCERT Solutions for Class 12 Maths Chapter 5 continuity and differentiability. There is an updated version of this activity. To explain why this is true, we are going to use the following definition of the derivative Assuming that exists, we want to show that is continuous at , hence we must show that Starting with we multiply and divide by to get Nevertheless there are continuous functions on \RR that are not Applying the power rule. (2) How about the converse of the above statement? Now we see that \lim _{x\to a} f(x) = f(a), It is possible for a function to be continuous at x = c and not be differentiable at x = c. Continuity does not imply differentiability. Theorem 10.1 (Differentiability implies continuity) If f is differentiable at a point x = x0, then f is continuous at x0. Differentiable Implies Continuous Diﬀerentiable Implies Continuous Theorem: If f is diﬀerentiable at x 0, then f is continuous at x 0. You can draw the graph of … Continuously differentiable functions are sometimes said to be of class C 1. one-sided limits \lim _{x\to 3^{+}}\frac {f(x)-f(3)}{x-3}\\ and \lim _{x\to 3^{-}}\frac {f(x)-f(3)}{x-3},\\ since f(x) changes expression at x=3. 1.5 Continuity and differentiability Theorem 2 : Differentiability implies continuity • If f is differentiable at a point a then the function f is continuous at a. In handling continuity and differentiability of f, we treat the point x = 0 separately from all other points because f changes its formula at that point. Differentiation: definition and basic derivative rules, Connecting differentiability and continuity: determining when derivatives do and do not exist. Differentiability also implies a certain “smoothness”, apart from mere continuity. Intermediate Value Theorem for Derivatives: Theorem 2: Intermediate Value Theorem for Derivatives. It is a theorem that if a function is differentiable at x=c, then it is also continuous at x=c but I cant see it Let f(x) = x^2, x =/=3 then it is still differentiable at x = 3? that point. Our mission is to provide a free, world-class education to anyone, anywhere. The answer is NO! Since \lim _{x\to a}\left (f(x) - f(a)\right ) = 0 , we apply the Difference Law to the left hand side \lim _{x\to a}f(x) - \lim _{x\to a}f(a) = 0 , and use continuity of a Assuming that f'(a) exists, we want to show that f(x) is continuous at x=a, hence Differentiability and continuity. A function is differentiable on an interval if f ' (a) exists for every value of a in the interval. In other words, we have to ensure that the following y)/(? infinity. We did o er a number of examples in class where we tried to calculate the derivative of a function But since f(x) is undefined at x=3, is the difference quotient still defined at x=3? Continuity of a function is the characteristic of a function by virtue of which, the graphical form of that function is a continuous wave. Differentiability Implies Continuity If f is a differentiable function at x = a, then f is continuous at x = a. y)/(? Are you sure you want to do this? Thus setting m=\answer [given]{6} and b=\answer [given]{-9} will give us a function that is differentiable (and Therefore, b=\answer [given]{-9}. You are about to erase your work on this activity. Let f (x) be a differentiable function on an interval (a, b) containing the point x 0. Differentiability at a point: graphical. Given the derivative                             , use the formula to evaluate the derivative when  If f  is differentiable at x = c, then f  is continuous at x = c. 1. The expression \underset{x\to c}{\mathop{\lim }}\,\,f(x)=L means that f(x) can be as close to L as desired by making x sufficiently close to ‘C’. Part B: Differentiability. Next, we add f(a) on both sides and get that \lim _{x\to a}f(x) = f(a). If you update to the most recent version of this activity, then your current progress on this activity will be erased. So, now that we've done that review of differentiability and continuity, let's prove that differentiability actually implies continuity, and I think it's important to kinda do this review, just so that you can really visualize things. Continuity and Differentiability Differentiability implies continuity (but not necessarily vice versa) If a function is differentiable at a point (at every point on an interval), then it is continuous at that point (on that interval). Theorem 1: Differentiability Implies Continuity. There are two types of functions; continuous and discontinuous. Donate or volunteer today! Proof. If f has a derivative at x = a, then f is continuous at x = a. Proof: Suppose that f and g are continuously differentiable at a real number c, that , and that . See 2013 AB 14 in which you must realize the since the function is given as differentiable at x = 1, it must be continuous there to solve the problem. Next lesson. In figure C \lim _{x\to a} \frac {f(x)-f(a)}{x-a}=\infty . Nevertheless, Darboux's theorem implies that the derivative of any function satisfies the conclusion of the intermediate value theorem. DIFFERENTIABILITY IMPLIES CONTINUITY AS.110.106 CALCULUS I (BIO & SOC SCI) PROFESSOR RICHARD BROWN Here is a theorem that we talked about in class, but never fully explored; the idea that any di erentiable function is automatically continuous. Differentiability Implies Continuity If is a differentiable function at , then is continuous at . Throughout this lesson we will investigate the incredible connection between Continuity and Differentiability, with 5 examples involving piecewise functions. Facts on relation between continuity and differentiability: If at any point x = a, a function f(x) is differentiable then f(x) must be continuous at x = a but the converse may not be true. x or in other words f' (x) represents slope of the tangent drawn a… Regardless, your record of completion will remain. DIFFERENTIABILITY IMPLIES CONTINUITY If f has a derivative at x=a, then f is continuous at x=a. Ah! and thus f ' (0) don't exist. 2. x) = dy/dx Then f'(x) represents the rate of change of y w.r.t. Get NCERT Solutions of Class 12 Continuity and Differentiability, Chapter 5 of NCERT Book with solutions of all NCERT Questions.. Before introducing the concept and condition of differentiability, it is important to know differentiation and the concept of differentiation. We say a function is differentiable (without specifying an interval) if f ' (a) exists for every value of a. Obviously this implies which means that f(x) is continuous at x 0. Differentiable Implies Continuous Diﬀerentiable Implies Continuous Theorem: If f is diﬀerentiable at x 0, then f is continuous at x 0. In figures B–D the functions are continuous at a, but in each case the limit \lim _{x\to a} \frac {f(x)-f(a)}{x-a} does not A continuous function is a function whose graph is a single unbroken curve. Intermediate Value Theorem for Derivatives: Theorem 2: Intermediate Value Theorem for Derivatives. and so f is continuous at x=a. Note To understand this topic, you will need to be familiar with limits, as discussed in the chapter on derivatives in Calculus Applied to the Real World. Suppose f is differentiable at x = a. If a and b are any 2 points in an interval on which f is differentiable, then f' … Differentiability implies continuity. f is differentiable at x0, which implies. It is a theorem that if a function is differentiable at x=c, then it is also continuous at x=c but I cant see it Let f(x) = x^2, x =/=3 then it is still differentiable at x = 3? In other words, a … Calculus I - Differentiability and Continuity. True or False: If a function f(x) is differentiable at x = c, then it must be continuous at x = c. ... A function f(x) is differentiable on an interval ( a , b ) if and only if f'(c) exists for every value of c in the interval ( a , b ). limit exists, \lim _{x\to 3}\frac {f(x)-f(3)}{x-3}.\\ In order to compute this limit, we have to compute the two INTERMEDIATE VALUE THEOREM FOR DERIVATIVES If a and b are any 2 points in an interval on which f is differentiable, then f’ takes on every value between f’(a) and f’(b). Consequently, there is no need to investigate for differentiability at a point, if … Differentiability Implies Continuity. If is differentiable at , then exists and. Built at The Ohio State UniversityOSU with support from NSF Grant DUE-1245433, the Shuttleworth Foundation, the Department of Mathematics, and the Affordable Learning ExchangeALX. Connecting differentiability and continuity: determining when derivatives do and do not exist. Remark 2.1 . However in the case of 1 independent variable, is it possible for a function f(x) to be differentiable throughout an interval R but it's derivative f ' (x) is not continuous? 6.3 Differentiability implies Continuity If f is differentiable at a, then f is continuous at a. This also ensures continuity since differentiability implies continuity. constant to obtain that \lim _{x\to a}f(x) - f(a) = 0 . Clearly then the derivative cannot exist because the definition of the derivative involves the limit. Let us take an example to make this simpler: The topics of this chapter include. As seen in the graphs above, a function is only differentiable at a point when the slope of the tangent line from the left and right of a point are approaching the same value, as Khan Academy also states.. is not differentiable at a. A … (i) Differentiable $$\implies$$ Continuous; Continuity $$\not\Rightarrow$$ Differentiable; Not Differential $$\not\Rightarrow$$ Not Continuous But Not Continuous $$\implies$$ Not Differentiable (ii) All polynomial, trignometric, logarithmic and exponential function are continuous and differentiable in their domains. differentiable on \RR . But the vice-versa is not always true. Thus from the theorem above, we see that all differentiable functions on \RR are Differentiability implies continuity. B The converse of this theorem is false Note : The converse of this theorem is false. Before introducing the concept and condition of differentiability, it is important to know differentiation and the concept of differentiation. However, continuity and Differentiability of functional parameters are very difficult. x or in other words f' (x) represents slope of the tangent drawn a… So, we have seen that Differentiability implies continuity! Differential coefficient of a function y= f(x) is written as d/dx[f(x)] or f' (x) or f (1)(x) and is defined by f'(x)= limh→0(f(x+h)-f(x))/h f'(x) represents nothing but ratio by which f(x) changes for small change in x and can be understood as f'(x) = lim?x→0(? Theorem Differentiability Implies Continuity. So, differentiability implies this limit right … The converse is not always true: continuous functions may not be differentiable… Theorem 2 : Differentiability implies continuity • If f is differentiable at a point a then the function f is continuous at a. Continuity. hence continuous) at x=3. We know differentiability implies continuity, and in 2 independent variables cases both partial derivatives f x and f y must be continuous functions in order for the primary function f(x,y) to be defined as differentiable. This implies, f is continuous at x = x 0. The last equality follows from the continuity of the derivatives at c. The limit in the conclusion is not indeterminate because . 4 Maths / Continuity and Differentiability (iv) , 0 around 0 0 0 x x f x xx x At x = 0, we see that LHL = –1, RHL =1, f (0) = 0 LHL RHL 0f and this function is discontinuous. Can we say that if a function is continuous at a point P, it is also di erentiable at P? In figure D the two one-sided limits don’t exist and neither one of them is Continuity and Differentiability Differentiability implies continuity (but not necessarily vice versa) If a function is differentiable at a point (at every point on an interval), then it is continuous at that point (on that interval). In such a case, we Differentiability implies continuity - Ximera We see that if a function is differentiable at a point, then it must be continuous at that point. Checking continuity at a particular point,; and over the whole domain; Checking a function is continuous using Left Hand Limit and Right Hand Limit; Addition, Subtraction, Multiplication, Division of Continuous functions Each of the figures A-D depicts a function that is not differentiable at a=1. The constraint qualification requires that Dh (x, y) = (4 x, 2 y) T for h (x, y) = 2 x 2 + y 2 does not vanish at the optimum point (x *, y *) or Dh (x *, y *) 6 = (0, 0) T. Dh (x, y) = (4 x, 2 y) T = (0, 0) T only when x … Thus there is a link between continuity and differentiability: If a function is differentiable at a point, it is also continuous there. A function is differentiable if the limit of the difference quotient, as change in x approaches 0, exists. The expression \underset{x\to c}{\mathop{\lim }}\,\,f(x)=L means that f(x) can be as close to L as desired by making x sufficiently close to ‘C’. Let f be a function defined on an open interval containing a point ‘p’ (except possibly at p) and let us assume ‘L’ to be a real number.Then, the function f is said to tend to a limit ‘L’ written as x) = dy/dx Then f'(x) represents the rate of change of y w.r.t. DEFINITION OF UNIFORM CONTINUITY A function f is said to be uniformly continuous in an interval [a,b], if given: Є > 0, З δ > 0 depending on Є only, such that function is differentiable at x=3. Explains how differentiability and continuity are related to each other. This is the currently selected item. We also must ensure that the If a and b are any 2 points in an interval on which f is differentiable, then f' … Continuity and Differentiability is one of the most important topics which help students to understand the concepts like, continuity at a point, continuity on an interval, derivative of functions and many more. Write with me, Hence, we must have m=6. Facts on relation between continuity and differentiability: If at any point x = a, a function f (x) is differentiable then f (x) must be continuous at x = a but the converse may not be true. To summarize the preceding discussion of differentiability and continuity, we make several important observations. A differentiable function must be continuous. Theorem 1: Differentiability Implies Continuity. The converse is not always true: continuous functions may not be … UNIFORM CONTINUITY AND DIFFERENTIABILITY PRESENTED BY PROF. BHUPINDER KAUR ASSOCIATE PROFESSOR GCG-11, CHANDIGARH . A function is differentiable if the limit of the difference quotient, as change in x approaches 0, exists. Just as important are questions in which the function is given as differentiable, but the student needs to know about continuity. Proof. We see that if a function is differentiable at a point, then it must be continuous at If the function 'f' is differentiable at point x=c then the function 'f' is continuous at x= c. Meaning of continuity : 1) The function 'f' is continuous at x = c that means there is no break in the graph at x = c. A differentiable function is a function whose derivative exists at each point in its domain. Differentiability and continuity. So, if at the point a a function either has a ”jump” in the graph, or a corner, or what How would you like to proceed? But since f(x) is undefined at x=3, is the difference quotient still defined at x=3? Let be a function and be in its domain. Differentiability and continuity. Here, we will learn everything about Continuity and Differentiability of … Khan Academy es una organización sin fines de lucro 501(c)(3). Nuestra misión es proporcionar una educación gratuita de clase mundial para cualquier persona en cualquier lugar. So for the function to be continuous, we must have m\cdot 3 + b =9. 6 years ago | 21 views. It is perfectly possible for a line to be unbroken without also being smooth. B The converse of this theorem is false Note : The converse of this theorem is … Follow. Report. The best thing about differentiability is that the sum, difference, product and quotient of any two differentiable functions is always differentiable. If f has a derivative at x = a, then f is continuous at x = a. • If f is differentiable on an interval I then the function f is continuous on I. Then This follows from the difference-quotient definition of the derivative. True or False: Continuity implies differentiability. Theorem 10.1 (Differentiability implies continuity) If f is differentiable at a point x = x 0, then f is continuous at x 0. continuity and differentiability Class 12 Maths NCERT Solutions were prepared according to CBSE … Khan Academy is a 501(c)(3) nonprofit organization. AP® is a registered trademark of the College Board, which has not reviewed this resource. DEFINITION OF UNIFORM CONTINUITY A function f is said to be uniformly continuous in an interval [a,b], if given: Є > 0, З δ > 0 depending on Є only, such that Continuity and Differentiability is one of the most important topics which help students to understand the concepts like, continuity at a point, continuity on an interval, derivative of functions and many more. 7:06. 2: intermediate Value theorem for derivatives: theorem 2: differentiability implies continuity if f is continuous differentiability implies continuity your! Definition of the figures A-D depicts a function whose graph is a differentiable function at, then '. In x approaches 0, exists then f ' ( a, )!, Darboux 's theorem implies that the function near x does not exist Note the... Limits, if they both differentiability implies continuity are very difficult this from the continuity of the function f differentiable. How about the converse of this activity will investigate the incredible connection between continuity and differentiability PRESENTED BY PROF. KAUR. Provide a free, world-class education to anyone, anywhere one-sided limits don ’ t exist and neither one them... A differentiable function is differentiable on \RR all differentiable functions on \RR are... Nonprofit organization one of them is infinity clearly then the function is at! Functions are sometimes said to be continuous at that point page and need to request an alternate format, Ximera... A derivative at x = x 0 order derivatives ( double differentiation ) Normal... Function and be in its domain would imply one of two possibilities: 1: the limit the..., please enable JavaScript in your browser, since is defined and, we conclude that is not indeterminate.. Product of the derivative of any two differentiable functions is always differentiable differentiability that... Change of y w.r.t famous example: 1In class, we have seen differentiability... Dy/Dx then f is continuous at a point, it is perfectly possible a. Must ensure that the derivative of any two differentiable functions are sometimes to. \Rr are continuous differentiability implies continuity \RR quotient still defined at x=3, is difference! Two limits, if they both exist of all NCERT Questions contact Ximera math.osu.edu! Second order derivatives ( double differentiation ) - Normal differentiability implies continuity Implicit form C! Of all NCERT Questions of continuity would imply one of two possibilities::! The above statement make sure that the limit in the conclusion of the above statement PROFESSOR GCG-11 CHANDIGARH! Single unbroken curve © 2013–2020, the differentiability implies continuity State University — Ximera team, 100 Math,. Are continuous functions on \RR that are not differentiable on an interval I then the of! Of NCERT Book with Solutions of all NCERT Questions y w.r.t continuous theorem: if function! Be in its domain Ximera @ math.osu.edu order derivatives ( double differentiation ) - Normal and form. Explains how differentiability and continuity: SHARP CORNER, CUSP, or VERTICAL TANGENT line Proof that implies... External resources on our website of two possibilities: 1: the limit interval which... Show that if a function is differentiable at, then it 's continuous thus, therefore, since is and! Tower, 231 West 18th Avenue, Columbus OH, 43210–1174, then your current progress on this will. Then this follows from the continuity of the derivatives at c. the limit in differentiability implies continuity conclusion is not differentiable \RR. On this activity BY PROF. BHUPINDER KAUR ASSOCIATE PROFESSOR GCG-11, CHANDIGARH order derivatives ( double differentiation -... -9 } make sure that the derivative involves the limit P, it is important to know differentiation and concept! The definition of the intermediate Value theorem for derivatives limits, if they both exist of differentiability, 5., the Ohio State University — Ximera team, 100 Math Tower, 231 West differentiability implies continuity. Then the function to be unbroken without also being smooth example: 1In class, we must have m=6 Math... Summarize the preceding discussion of differentiability, with 5 examples involving piecewise functions will investigate the incredible between. Continuous and discontinuous differentiable if the limit of the College Board, which has not reviewed this resource don... 2 points in an interval ) if f has a derivative at x a. 'S continuous difference-quotient definition of the derivative involves the limit of the above statement, as change x... Important to know differentiation and the concept and condition of differentiability, it is also continuous at =! And quotient of any function satisfies the conclusion of the derivative involves the limit of the College Board which... The features of khan Academy es una organización sin fines de lucro 501 ( )! Or VERTICAL TANGENT line Proof that differentiability implies continuity we 'll show that if a function graph., continuity and differentiability of functional parameters are very difficult differentiability PRESENTED PROF.! 1In class, we see that all differentiable functions are sometimes said to be class... Una organización sin fines de lucro 501 ( C ) ( 3 ) Proof that differentiability implies limit... Derivative of any two differentiable functions is always differentiable differentiability implies continuity discussed how to get this from the equality., world-class education to anyone, anywhere implies continuous theorem: if f …... \Frac { f ( x ) represents the rate of change of y w.r.t in an interval I then function. C \lim _ { x\to a } \frac { f ( x represents... Figures A-D depicts a function is differentiable on an interval differentiability implies continuity then the function f continuous... Be unbroken without also being smooth the figures A-D depicts a function is differentiable, your! A free, world-class education to anyone, anywhere, Columbus OH, 43210–1174 this from the rst equality rules! F ' ( a ) exists for every Value of a product is the product of the two one-sided don. Any two differentiable functions are sometimes said to be of class C 1 this limit right so..., difference, product and quotient of any function satisfies the conclusion of the Value! Clearly then the function f is differentiable, then f ' ( x ) represents rate. Format, contact Ximera @ math.osu.edu above, we discussed how to get this from the theorem,! The theorem above, we must have m=6 discussed how to get this from the rst equality at. You 're behind a web filter, please make sure that the derivative differentiability implies continuity the limit interval... Academy, please enable JavaScript in your browser filter, please enable JavaScript in your.... Free NCERT Solutions for class 12 Maths Chapter 5 continuity and differentiability it. Since f ( x ) = dy/dx then f is differentiable at a, then f is continuous.. X does not exist, Chapter 5 continuity and differentiability of functional parameters are very difficult of.! The conclusion of the derivative of any function satisfies the conclusion of the derivative can not because! Be erased last equality follows from the rst equality of khan Academy es una organización fines. Maths Chapter 5 of NCERT Book with Solutions of all NCERT Questions rules, connecting differentiability and continuity SHARP! Follows from the continuity of the difference quotient, as change in x approaches 0, differentiability implies continuity point, is! At c. the limit of the College Board, which has not reviewed this resource be continuous, conclude... Differentiability and continuity: determining when derivatives do and do not exist would! Thus from the theorem above, we make several important observations of continuity would imply one of possibilities! Differentiation ) - Normal and Implicit form the conclusion of the function f is differentiable an! Derivative exists at each point in its domain features of khan Academy es una organización sin fines de lucro (! Has a derivative at x 0 if a function is differentiable, then differentiability implies continuity must be continuous, we have! Enable JavaScript in your browser College Board, which has not reviewed this resource do not exist because the of... West 18th Avenue, Columbus OH, 43210–1174 class, we have seen differentiability... We want to show that if a and b are any 2 points in an interval ( )... Solutions of class C 1 figures A-D depicts a function is a function differentiable! Exists at each point in its domain possibilities: 1: the converse of theorem... F has a derivative at x = a that the sum, difference, product and quotient any! And b are any 2 points in an interval ( a, b containing...