The classical interior uniqueness theorem for holomorphic (that is, single-valued analytic) functions on D states that if two holomorphic functions f(z) and g(z) in D coincide on some set E⊂D containing at least one limit point in D, then f(z)≡g(z) everywhere in D.

Are holomorphic functions entire?

A holomorphic function whose domain is the whole complex plane is called an entire function. The phrase "holomorphic at a point z₀" means not just differentiable at z₀, but differentiable everywhere within some neighbourhood of z₀ in the complex plane.

Are all analytic functions differentiable?

Any analytic function is smooth, that is, infinitely differentiable. The converse is not true for real functions; in fact, in a certain sense, the real analytic functions are sparse compared to all real infinitely differentiable functions.