If f is complex differentiable at every point z0 in an open set U, we say that f is holomorphic on U. ... A simple converse is that if u and v have continuous first partial derivatives and satisfy the Cauchy–Riemann equations, then f is holomorphic.
Is holomorphic function continuous?
The derivative of a holomorphic function is always continuous. This similar result doesn't hold in the context of real analysis: there are some real-valued functions of a real variable that are differentiable and whose derivative is not continous1.
Does analytic imply continuous?
And if a function is analytic does this mean it is continuous? Yes. Every analytic function has the property of being infinitely differentiable. Since the derivative is defined and continuous, the function is continuous everywhere.
