Allied & ICAR Exam Bank • ALLIED_2635 MCQ • 1 Mark

Section 1: Engineering Mathematics

Calculus

Subtopic: Limit, continuity and differentiability

Question Statement:

If a function $f(x)$ is differentiable at $x = c$, then:

Options:

  • (A) $f(x)$ must be continuous at $x = c$
  • (B) $f(x)$ may or may not be continuous at $x = c$
  • (C) $f'(c)$ must be strictly positive
  • (D) $f''(c)$ must exist

Answer Key:

A

Step-by-Step Solution & Explanation

Differentiability strictly implies continuity: if $f'(c)$ exists, then $\lim_{x \to c} [f(x) - f(c)] = \lim_{x \to c} \frac{f(x)-f(c)}{x-c} (x-c) = f'(c) \cdot 0 = 0$.
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