JEE Advanced · 2026 · 17 May 2026 · Paper 1 · Continuity and Differentiability

(A) FALSE. Counterexample: $f(x)=\frac{1}{x}\sin (1/x)$ for $x\ne 0$, $f(0)=0$. Then $g(x)=\sin (1/x)$ for $x\ne 0$, $g(0)=0$, which is not continuous at 0.

(B) TRUE. If $f$ is continuous at 0, then $f(0)$ is finite. We have

which exists and is finite, so $g$ is differentiable at 0.

(C) FALSE. If $g$ is differentiable at 0, then $g^{\prime }(0)={\lim }_{h\to 0}f(h)$ exists, but this doesn't guarantee $f(0)$ equals this limit, so $f$ need not be continuous at 0.

(D) TRUE. As shown in (C), if $g$ is differentiable at 0, then ${\lim }_{x\to 0}f(x)$ exists.