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NDA 2020 Maths · Mathematics · Continuity and Differentiability · easy

Every continuous function is differentiable. 2. Every differentiable function is continuous.

Single correct option

Consider the following statements:
1. Every continuous function is differentiable.
2. Every differentiable function is continuous.
Which of the statements is/are correct?
  1. A. 1 only
  2. B. 2 only
  3. C. Both 1 and 2
  4. D. Neither 1 nor 2
Show answer and solution

Answer: (B) 2 only

Differentiability at a point implies continuity there, since f(x)−f(a)=f(x)−f(a)x−a (x−a)→f′(a)⋅0=0f(x)-f(a)=\dfrac{f(x)-f(a)}{x-a}\,(x-a)\to f'(a)\cdot0=0. So statement 2 is correct. The converse is false: f(x)=∣x∣f(x)=|x| is continuous at 00 but has a corner there, so it is not differentiable. Hence statement 1 is incorrect. Remember the direction: differentiable ⇒\Rightarrow continuous, not the other way round.
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Original question written in the NDA 2020 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake