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

Let $a$ be any point

Now, at $x=a,\underset{x\to a}{\lim }f\left(x\right)=\underset{h\to 0}{\lim }f\left(a+h\right)$

$=\underset{h\to 0}{\lim }f\left(a\right)+\underset{h\to 0}{\lim }f\left(h\right)$

$=f\left(a\right)+f\left(0\right)=f\left(a+0\right)=f\left(a\right)$

$\therefore f\left(x\right)$ is continuous at $x=a$, where $a$ is any arbitrary point.

Hence, $f\left(x\right)$ is continuous for all $x$.