$\displaystyle \lim_{x\to 0}\left|\frac{\sin x}{x}\right| = 1 = f(0)$

$\Rightarrow f(x)$ is continuous.

Now,

$\displaystyle \frac{d}{dx}\left(\frac{\sin x}{x}\right) = \frac{x\cos x-\sin x}{x^2} $

$\Rightarrow f^{\prime }(x)=0$

$\Rightarrow \tan x=x$

There are $3$ solutions of $\tan x=x$ in the interval $(-2\pi ,2\pi )$.

Also, $f^{\prime }(x)$ does not exist at $x=-\pi ,\pi$.

Therefore, total number of points $=3+2=5$.

$\boxed{5}$