MCQ
Let $f(x)\,\, = \,\left\{ {\begin{array}{*{20}{c}}
{x\,\sin \,\frac{1}{x}\,\sin \,\left( {\frac{1}{{x\,\sin \,\frac{1}{x}}}} \right)\,,\,\,x\, \ne \,0}\\
{0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\, = \,0\,}
\end{array}} \right.{\mkern 1mu} $ then $f(x)$ is
{x\,\sin \,\frac{1}{x}\,\sin \,\left( {\frac{1}{{x\,\sin \,\frac{1}{x}}}} \right)\,,\,\,x\, \ne \,0}\\
{0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\, = \,0\,}
\end{array}} \right.{\mkern 1mu} $ then $f(x)$ is
- Aboth continuous and differentiable at $x = 0$
- ✓
- Cneither continuous nor differentiable at $x = 0$
- D$f '(0^-)$ exists.