Solution: The function provided is

$f(x)=\left\{\begin{array}{lll}2 x, & \text { if } & x<0 \\ 0, & \text { if } & 0 \leq x \leq 1 \\ 4 x, & \text { if } & x>1\end{array}\right.$

At $x=0$,

Left Hand Limit $=\lim _{x \rightarrow 0} 2 x=0$

and Right Hand Limit $=\lim _{x \rightarrow 0^{-}}(0)=0$

Since L.H.L.$\neq$ R.H.L.

As a result, $f(x)$ is continuous at $x=0$

At $x=1$,

Left Hand Limit $=\lim _{x \rightarrow 1}(0)=0$

and Right Hand Limit $=\lim _{x \rightarrow 1}(4 x)=4$

Since, L.H.L. $\neq$ R.H.L.

As a result, $f(x)$ is discontinuous at $x=1$

When the value of $x<0$,

$f(x)$ is a polynomial function and is continuous for all $x<0$

When the value of $x>1, f(x)=4 x$

Being a polynomial function it is continuous for all $x>1$.

As a result, $x=1$ is a point of discontinuity.