GATE 2021 CH – Question 26

Engineering Mathematics · Calculus: Limits, continuity, differentiability, Taylor series and mean value theorem · 1 mark · Multiple select

For the function $f(x)=\begin{cases}-x,&x<0\\x^2,&x\ge0\end{cases}$, the CORRECT statement(s) is/are

$f(x)$ is continuous at $x=1$
$f(x)$ is differentiable at $x=1$
$f(x)$ is continuous at $x=0$
$f(x)$ is differentiable at $x=0$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (A) $f(x)$ is continuous at $x=1$; (B) $f(x)$ is differentiable at $x=1$; (C) $f(x)$ is continuous at $x=0$

Explanation

Near $x=1$, the function is the polynomial $x^2$, so it is both continuous and differentiable: **A and B are true**.

At zero, the left limit of $-x$ and right limit of $x^2$ are both zero, equal to $f(0)$: **C is true**. However, the one-sided derivatives are $-1$ and 0, respectively. Since they differ, $f$ is not differentiable at zero: **D is false**.

**Answer: A, B and C.**