Question

Examine the continuity of the function $f\left(x\right)$ at $x=0,$ where
$f\left(x\right)=x\sin \frac{1}{x},\text{for}x\ne 0$
$=0,\text{for}x=0$

Answer 1

$f\left(x\right)=x\sin \frac{1}{x}$ for $x\ne 0$

$=0$ for $x=0$

Solution:

$\underset{x\to 0}{lim}f\left(x\right)=\underset{x\to 0}{lim}x\sin \frac{1}{x}$ $=0$

at $x=0$ $f\left(0\right)=0$

$\underset{x\to 0}{lim}f\left(x\right)=f\left(0\right)=0$

So the given$f\left(x\right)$ is continuous at $x=0$.