Let f,g:RR be defined by f(x)=casesx1x, & x00, & x=0cases, and g(x)=xf(x).

Question

Let $f,g:\mathbb{R}	o\mathbb{R}$ be defined by $f(x)=\begin{cases}x\sin\frac1x, & x
e0\0, & x=0\end{cases}$, and $g(x)=xf(x)$.

Statement I: $f$ is a continuous function at $x=0$.

Statement II: $g$ is a differentiable function at $x=0$.

Accepted Answer

For $f$: LHL $=$ RHL $=f(0)=0$ (bounded $\sin$ term times a vanishing factor), so $f$ is continuous at $0$ --- Statement I true.

$g(x)=x^2\sin\frac1x$ (for $x
e0$), $0$ at $x=0$. LHD $=$ RHD $=0$, so $g$ is differentiable at $0$ --- Statement II true.

Solutions · 1