Chapter 2: Q. 96 (page 186)

Prove that if a function $f$ is differentiable at $x = c$ , then f is continuous at $x = c$ .
(a) We are given that f is differentiable at $x = c$ . Use the alternative definition of the derivative to write down what that statement means.
(b) We want to show that f is continuous at $x = c$ . Use the definition of continuity to show that this statement is equivalent to the statement $\lim_{x \to c} (f (x) - f (c)) = 0$ .
(c) Now use part (a) to show that $\lim_{x \to c} (f (x) - f (c)) = 0$ .
(Hint: Multiply ( $f (x) - f (c) b y \frac{x - c}{x - c}$ and use the product rule for limits.)

Short Answer

Expert verified

Ans:

Part (a). $\lim_{x \to c} \frac{f (x) - f (c)}{x - c} = f (c) i t i s c o n t i n u o u s a t x = c$

Part (b). $\lim_{x \to c^{-}} f (x) = \lim_{x \to c^{+}} f (x) = f (c)$

Part (c). $\lim_{x \to c} [f (x) - f (c)] = 0$

Step by step solution

Step 1. Given information:

function $f$ is differentiable at $x = c$

Step 2. Solving Part (a):

Since $f$ is differentiable at $x = c$

$\lim_{x \to c} \frac{f (x) - f (c)}{x - c} = f' (c) \frac{\lim_{x \to c} (f (x) - f (c))}{\lim_{x \to c} (x - c)} = f' (c) u \sin g d i v i s i o n r u l e \lim_{x \to c} (f (x) - f (c)) = f' (c) [\lim_{x \to c} (x - c)] \lim_{x \to c} f (x) - \lim_{x \to c} f (c) = f' (c) \cdot 0 \lim_{x \to c} f (x) = f (c) H e n c e i t i s c o n t i n u o u s a t x = c$

Step 3. Solving Part (b):

A function is said to be continuous at $\lim_{x \to c^{-}} f (x) = \lim_{x \to c^{+}} f (x) = f (c)$

The limit of the function exists at $x = c$

$\lim_{x \to c} f (x) = f (c) \lim_{x \to c} f (x) = \lim_{x \to c} f (c) \lim_{x \to c} f (x) - \lim_{x \to c} f (c) = 0 \lim_{x \to c} [f (x) - f (c)] = 0 H e n c e b o t h s t a t e m e n t s a r e e q u i v a l e n t$

Step 4. Solving Part (c):

$\lim_{x \to c} [f (x) - f (c)] = \lim_{x \to c} [[f (x) - f (c)] (\frac{x - c}{x - c})] = \lim_{x \to c} \frac{[f (x) - f (c)]}{x - c} \cdot \lim_{x \to c} (x - c) = f' (c) \cdot \lim_{x \to c} (x - c) = f' (c) \cdot (c - c) \lim_{x \to c} [f (x) - f (c)] = 0$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Solving Part (a):

Step 3. Solving Part (b):

Step 4. Solving Part (c):

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Theoretical and Mathematical Physics

Discrete Mathematics

Decision Maths

Applied Mathematics

Logic and Functions

Study anywhere. Anytime. Across all devices.