one definition of a convex function is that its epigraph

    { (x, y) : f(x) <= y }

is a convex set. Another is that the line segment between two points on the graph lies above the graph, i.e.

    (1 - t) f(x) + t f(y) >= f((1 - t) x + t y) for all 0 <= t <= 1

known as Jensen's inequality.

Convex function must be defined on a convex domain (no holes) and continuous at everywhere except the boundary of the domain.