What are the necessary and sufficient conditions for an epigraph of a function to be a polyhedron?

Question

Should the function be a] Piecewise affine or b] Convex and piecewise affine?

I'm doing the MOOC on Convex Optimization by Stephen Boyd and came across this question in the exercises. The answer which is given there is b] but I think a] suffices.

Answer 1

Hint: According to your definition, a polyhedron is always convex. What about the epigraph of a function?