Convex function and curves

Question

Is there diffrence between convex function and convex curve i.e Is it true that A function is convex iff its curve is convex .or they are not related with each other . It will better if there is mathematic proof if its true or counter example if its false

Answer 1

A function from a convex subset $X$ of a real vector space $E$, to $\mathbb{R}$, is convex iff its epigraph, i.e. the set of points above the function curve or surface, is convex in $E \times \mathbb{R}$.

So yes, the two are equivalent.

Of course if the function's domain $X$ is not convex, the epigraph cannot be convex.