## Help with a simple proof

Let f be continuous at a. Prove that |f| is continuous at a.

Proof: If $f(x)\geq 0$ then the statement is true by hypothesis.
If $f(x)<0$ then $|f(x)|=-f(x)$.
So we want to show that $\forall \epsilon >0 \exists \delta >0$ such that $|f(x)-f(a)|<\epsilon$ when $|x-a|<\delta$.
$||f(x)|-f(a)|=|f(x)+f(a)|$ because f(x)<0.
...
Now what can I say?