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). Thus, a discontinuous convex function is unbounded on any interior interval and is not measurable.

Can a discontinuous function be concave or convex?

It is convex but not strictly convex. Convex (or concave) funtions are continuous over the relative interior of their domain. A discontinuous function could not be convex nor concave on all of its domain - but it can of course be piecewise convex (or concave) over it's continuity regions.

Do convex functions have to be continuous?

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.