An elementary proof for functions on open subsets of the plane is as follows (by a simple reduction, the general case for the theorem of Schwarz easily reduces to the planar case). Let f ( x , y ) be a differentiable function on an open rectangle Ω Ω containing a point ( a , b ) and suppose that d f is continuous with continuous ∂ ∂ x ∂ ∂ y f and ∂ ∂ y ∂ ∂ x f over Ω Ω . Define