Osgood's lemma
In mathematics, Osgood's lemma, introduced by, is a proposition in complex analysis. It states that a continuous function of several complex variables that is holomorphic in each variable separately is holomorphic. The assumption that the function is continuous can be dropped, but that form of the lemma is much harder to prove and is known as Hartogs' theorem.
There is no analogue of this result for real variables. If it is assumed that a function is globally continuous and separately differentiable on each variable, it is not true that will necessarily be differentiable. A counterexample in two dimensions is given by
If in addition it is defined that, this function is everywhere continuous and has well-defined partial derivatives in and everywhere, but is not differentiable at the origin.