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?