I received a request for some hints on Theorem 2.15 and I figured it would be a nice to post them here for everyone to see.
You need to pick one of our four definitions for continuity and probably stick with it throughout the proof. The proof I'm thinking of uses the definition involving open intervals. Here are some hints to get you started:
Let $S=(a,b)$ be an open interval containing $h(p)$. Since $f(p)=h(p)=g(p)$, $f(p)$ and $g(p)$ are also contained in $S$. We are assuming that $f$ and $g$ are continuous. What does our definition of continuity involving open sets tell us in this case? You should say something about two open intervals on the $x$-axis. Now, using the two intervals that you have associated to $f$ and $g$, you need to construct an open interval $T_h$ on the $x$-axis such that if $t\in T_h$, then $h(t)\in S$. At some point, you'll need to encounter a statement like: $a<f(t)\leq h(t)\leq g(t)<b$.