The convex hull of a function.

Question

I'm doing this exercise which gives the following definition of the convex hull of a function.

Doesn't this definition make $g(x) = f(x)$?

Answer 1

No: for instance, if $f(x)=\begin{cases}1&\text{if }0\le x\le 1\\ 0&\text{if }x<0\lor x>1\end{cases}$, then $g=0$, because $\operatorname{conv}\operatorname{epi} f=\Bbb R\times [0,\infty)$.