You can take for granted that the difference of continuous functions is continuous and you can use $\epsilon$ - $\delta$ definition of continuity.
i) If $f \colon \Bbb R \rightarrow \Bbb R $, $g \colon \Bbb R \rightarrow \Bbb R $, where $f$ and $g$ are continuous and $f(x_0) >g(x_0)$ for some $x_0$ element in $\Bbb R$, then there exists a $\delta$ such that $f(x)$-$g(x)$ > 1/2 $(f(x_0)$ - $g(x_0))$ for all $x$ element in ($x_0$ - $\delta$, $x_0$ + $\delta$).
ii) use i) to show by contradiction that if $f \colon \Bbb R \rightarrow \Bbb R $, $g \colon \Bbb R \rightarrow \Bbb R $, where $f$ and $g$ are continuous, and $f(x)$=$g(x)$ a.e. then $f$=$g$.
My TA told me to not look at this problem again after I asked him because we will not be covering this topic. I was still wondering what the proof will look like though. It seems like a good proof to look at for something I might learn later on in another self study class.
OK. I understand part i) of the question. How would be prove part ii)?