The task is to prove that $f-g$ and $|f|$ are continuous in a given that two real functions $f$ and $g$ are both continuous in $a \in \mathbb R$.
Proof of $f-g$:
First of all one has to prove that $g$ is continuous and that implies that $-g$ is continuous. After that one needs to use the proof of the sum of two functions $f$ and $g$ so that $f+(-g)$. I've already written that proof in here [1]
So I don't know how to prove that g is continuous unless it is directly from the $\epsilon$-$\delta$-definition but that I know wouldn't be sufficient.
Proof of $|f|$:
For this proof one needs to use the triangle inequality.
$(\forall \epsilon \gt 0)(\exists \delta\gt 0)(\forall x \in \mathbb R)(|x-a|\lt \delta \Longrightarrow ||f(x) -f(a)||\lt \epsilon$
$\Longrightarrow ||f(x)- f(a)|| \le ||f(x)| -|f(a)|| \lt \epsilon$
This is the best I've got. Can you please help me proving the last two statements.
[1] : (Prove the continuity of $f-g$ and $|f|$ when $f$, $g$ are continuous.)