It can be shown that if a function of a complex variable is differentiable (in some neighborhood, not just at a point), then it is analytic, which immediately implies it has derivatives of all orders, which, finally, means that its deerivative is differentiable, and so continuous.

If, by F(z)= \int f(z)dz, we mean the function whose derivative, in some region, is 0, the F musthavea derivative, which implies it is analytic, which implies it derivative, f(z), is continuous.