Consider a function f defined on a real interval I⊂R. f is called convex if: ∀x,y∈I ∀λ∈[0,1]: f((1−λ)x+λy)≤(1−λ)f(x)+λf(y)
Suppose that I is a closed interval: I=[a,b] with a<b. For a<s<t<u<b one can prove that: f(t)−f(s)t−s≤f(u)−f(s)u−s≤f(u)−f(t)u−t. It follows from those relations that f has left-hand and right-hand derivatives at each point of the interior of I. And therefore that f is continuous at each point of the interior of I.
Is a convex function defined on an interval I continuous at all points of the interval? That might not be the case and a simple example is the function: f:[0,1]⟶Rx⟼0 for x∈(0,1)x⟼1 else
It can be easily verified that f is convex. However, f is not continuous at 0 and 1.