Any holomorphic function on a compact Riemann surface is necessarily constant. , so it is constant, by the maximum modulus principle.

Are all holomorphic functions continuous?

A function which is differentiable at a point in any usual sense of the word (including holomorphic, which is, after all, another name for complex differentiability) will be continuous at that point.

What makes a function holomorphic?

In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space Cⁿ. The existence of a complex derivative in a neighbourhood is a very strong condition: it implies that a ...