All measurable convex functions on open intervals are continuous. There exist convex functions which are not continuous, but they are very irregular: If a function f is convex on the interval (a,b) and is bounded from above on some interval lying inside (a,b), it is continuous on (a,b).

Are all continuous functions convex?

Since in general convex functions are not continuous nor are they necessarily continuous when defined on open sets in topological vector spaces. ... But every convex function on the reals is lower semicontinuous on the relative interior of its effective domain, which equals the domain of definition in this case.

Which functions are convex?

A differentiable function of one variable is convex on an interval if and only if its derivative is monotonically non-decreasing on that interval. If a function is differentiable and convex then it is also continuously differentiable. for all x and y in the interval. is strictly convex.