GATE 2026 IN – Question 12

Engineering Mathematics · Calculus and vector calculus · 1 mark · Multiple choice

Which one among the following statements is true for the function $f(x)=x|x|$?

$f(x)$ is discontinuous but not differentiable at $x=0$.
$f(x)$ is discontinuous and differentiable at $x=0$.
$f(x)$ is continuous but not differentiable at $x=0$.
$f(x)$ is continuous and differentiable at $x=0$.

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Correct answer: (D) $f(x)$ is continuous and differentiable at $x=0$.

Explanation

As $x\to0$, $|f(x)|=x^2\to0=f(0)$, so $f$ is continuous at zero.

Using the derivative definition, $f\prime(0)=\lim_{h\to0}[f(h)-f(0)]/h=\lim_{h\to0}|h|=0$. Thus it is also differentiable at zero. Both properties hold: **D**.