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

Proof: If $\displaystyle f(x)\geq 0$ then the statement is true by hypothesis.

If $\displaystyle f(x)<0$ then $\displaystyle |f(x)|=-f(x)$.

So we want to show that $\displaystyle \forall \epsilon >0 \exists \delta >0$ such that $\displaystyle |f(x)-f(a)|<\epsilon$ when $\displaystyle |x-a|<\delta$.

$\displaystyle ||f(x)|-f(a)|=|f(x)+f(a)|$ because f(x)<0.

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Now what can I say?