Page "Holomorphic function" Paragraph 12

If continuity is not a given, the converse is not necessarily true.

A simple converse is that if u and v have continuous first partial derivatives and satisfy the Cauchy – Riemann equations, then ƒ is holomorphic.

A more satisfying converse, which is much harder to prove, is the Looman – Menchoff theorem: if ƒ is continuous, u and v have first partial derivatives ( but not necessarily continuous ), and they satisfy the Cauchy – Riemann equations, then ƒ is holomorphic.