Osgood's lemma

In mathematics, Osgood's lemma, introduced by William Fogg Osgood (1899), 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 we assume that a function [math]\displaystyle{ f:\mathbb{R}^n\to\mathbb{R} }[/math] is globally continuous and separately differentiable on each variable (all partial derivatives exist everywhere), it is not true that [math]\displaystyle{ f }[/math] will necessarily be differentiable. A counterexample in two dimensions is given by

[math]\displaystyle{ f(x,y)=\dfrac{2x^2y+y^3}{x^2+y^2}. }[/math]

If in addition we define [math]\displaystyle{ f(0,0)=0 }[/math], this function is everywhere continuous and has well-defined partial derivatives in [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] everywhere (also at the origin), but is not differentiable at the origin.

References

• Osgood, William F. (1899), "Note über analytische Functionen mehrerer Veränderlichen", Mathematische Annalen (Springer Berlin / Heidelberg) 52: 462–464, doi:10.1007/BF01476172, ISSN 0025-5831, https://zenodo.org/record/1672740/files/article.pdf 
• Gunning, Robert Clifford; Rossi, Hugo (2009). Analytic Functions of Several Complex Variables. ISBN 978-0-8218-2165-7. https://books.google.com/books?id=wsqFAwAAQBAJ&pg=PA2.