In mathematical analysis, Schwarz's theorem (or Clairaut's theorem on equality of mixed partials) named after Alexis Clairaut and Hermann Schwarz, states that for a function f : : Ω Ω → → R defined on a set Ω Ω ⊂ ⊂ R n , if p ∈ ∈ R n is a point such that some neighborhood of p is contained in Ω Ω and f has continuous second partial derivatives on that neighborhood of p , then for all i and j in { 1 , 2 … … , n } ,