Show that $|f(x)|$ is continuous at $a$ 
Suppose that $f(x)$ is continuous at $a$. Show that the function $|f(x)|$ is continuous at $a$.

Proof:
Since $f(x)$ is continuous at $a$ then, $$\lim_{x\to a}f(x)=f(a)$$
Show that $|f(x)|$ is continouos at $a$ $$\lim_{x\to a}|f(x)|=...$$
From here I can not figure a way to finish the proof. In my head $|f(x)|$ might not be continuous a $a$, such as if $f(a)$ is negative. Then $|f(a)|$ would be positive. Any help would be appreciated! Preferably relating to the Basic Limit Theorems of continuity if possible.

First, show that the function $g(x)=|x|$ is continuous on reals, then use the fact that the composition of two continuous functions is also continuous ($|f(x)|= g \circ f (x)$).
