The GATE Grind

GATE 2016 EC – Question 13

Engineering Mathematics · Calculus · 1 mark · Multiple choice

Given the following statements about a function $f : \mathbb{R} \rightarrow \mathbb{R}$, select the right option:

P: If $f(x)$ is continuous at $x = x_0$, then it is also differentiable at $x = x_0$.

Q: If $f(x)$ is continuous at $x = x_0$, then it may not be differentiable at $x = x_0$.

R: If $f(x)$ is differentiable at $x = x_0$, then it is also continuous at $x = x_0$.

  1. P is true, Q is false, R is false
  2. P is false, Q is true, R is true
  3. P is false, Q is true, R is false
  4. P is true, Q is false, R is true

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (B) P is false, Q is true, R is true

Explanation

Continuity does not guarantee differentiability. For example, $|x|$ is continuous at 0 but has a corner there, so P is false and Q is true. Differentiability does imply continuity, so R is true.