Exercise · ZW-DIFFERENTIABILITY-AND-CONTINUITY-007 · ★★☆ Intermediate
In the proof that differentiable implies continuous, which identity is used for \(x\ne c\)?
Choose one option.
Show answer
- D. \(f(x)-f(c)=\frac{f(x)-f(c)}{x-c}\,(x-c)\)
Explanation
As \(x\to c\), the quotient tends to \(f'(c)\) and \(x-c\to0\); the product tends to \(0\), that is, \(f(x)\to f(c)\).
Review the concept: Differentiability and continuity
If a function \(f\) is differentiable at a point \(c\), then \(f\) is continuous at \(c\). The converse is false.
\[f'(c)\text{ exists}\ \Longrightarrow\ \lim_{x\to c}f(x)=f(c)\]
More practice problems: differentiability and continuity
- ZW-DIFFERENTIABILITY-AND-CONTINUITY-001 Which statement is true for a function f and a point c?
- ZW-DIFFERENTIABILITY-AND-CONTINUITY-002 What happens to f(x)=|x| at c=0?
- ZW-DIFFERENTIABILITY-AND-CONTINUITY-003 Let f(x)=x^1/3. What can be said at c=0?
- ZW-DIFFERENTIABILITY-AND-CONTINUITY-004 Let f(x)=x^2 if x≤1 and f(x)=ax+b if x>1. If f is differentiable at 1, what is a+b?
- ZW-DIFFERENTIABILITY-AND-CONTINUITY-005 Let f(x)=x^2sin (1/x) for xne0 and f(0)=0. What is f'(0)?
- ZW-DIFFERENTIABILITY-AND-CONTINUITY-006 If f is not continuous at c, what follows?
- ZW-DIFFERENTIABILITY-AND-CONTINUITY-008 Let f(x)=1 if x≥0 and f(x)=0 if x<0. Is f differentiable at 0?
- ZW-DIFFERENTIABILITY-AND-CONTINUITY-009 Let f(x)=xsin (1/x) for xne0 and f(0)=0. Which statement is true at 0?
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