Question

Show that the function $f\left(x\right)$ defined as $f\left(x\right)=x\cos \frac{1}{x},x\ne 0=0,x=0$
is continuous at $x=0$.

Answer 1

Given the function $f\left(x\right)$ will be continuous if $\underset{x\to 0}{lim}f\left(x\right)=f\left(0\right)$.

Now, we've,

$\left|\cos \frac{1}{x}\right|\le 1$, $\forall x\in R-\left\{0\right\}$

or, $\left|x\cos \frac{1}{x}\right|\le |x|$

Then, $\underset{x\to 0}{lim}x\cos \frac{1}{x}=0$ [ By Sandwich property]

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

So $f$ is continuous at $x=0$.