Question

If $f\left(x\right)=\{\begin{array}{ccc}{x}^{2}sin\frac{1}{x},& when& x\ne 0\\ 0& when& x=0\end{array}$, then

• A. f (0 + 0) = 1
• B. f (0 - 0) = 1
• C. f is continuous at x = 0
• D. None of these

Answer 1

The correct option is C f is continuous at x = 0

$li{m}_{x\to 0^{+}}f\left(x\right)=x^{2}sin\frac{1}{x},but-1\le sin\frac{1}{x}\le 1$ and $x\to 0$
Therefore, $li{m}_{x\to 0^{+}}f\left(x\right)=0={lim}_{x\to 0^{-}}f\left(x\right)=f\left(0\right)$
Hence f(x)is continuous at x = 0