I have to prove the following: Show that if $f : (a, b) →\mathbb R$ is continuous then $|f|$ is continuous on $(a, b)$ as well. 

So I want to show that $lim\_{x \rightarrow x\_0} |f(x)| = |f(x\_0)|$. It's easy to show this for $x\_0$ s.t. $f(x\_0) < 0$ or $f(x\_0) > 0$ because we know, for example, that if $lim\_{x \rightarrow x\_0} f(x) = f(x\_0) > 0$ then there must exist a $\delta$ s.t. $f(x) < 0$ $\forall |x-x\_0|<\delta$ so we can say $lim\_{x \rightarrow x\_0} |f(x)| = lim\_{x \rightarrow x\_0} f(x) = f(x\_0) = |f(x\_0)|$.
However I'm not sure how to show it for $x\_0$ s.t. $f(x\_0) = 0$. The way I tried to show it $x\_0$ s.t. $f(x\_0) = 0$ was by saying that if $f(x\_0) = 0$ then $lim\_{x \rightarrow x\_0} f(x) = f(x\_0) = 0 =|f(x\_0)|$ so we must have $lim\_{x \rightarrow x\_0} |f(x)| = |f(x\_0)|$. But this to me just seemed trivially true and I wasn't really sure if my justification was a justification at all.