In _Calculus and Analytic Geometry_ 2nd ed.(1953), Thomas provides:
Theorem. If the function w=f(x,y) together with the partial derivatives f_x, f_y, f_xy and f_yx are continuous, then f_xy = f_yx.
Both Thomas and Anton (1980) provide rather long-winded proofs of this theorem. These proofs involved geometric arguments, auxiliary functions, the mean-value theorem, epsilon error variables, a proliferation of symbols, and a generous helping of obscurity.
Starting from the definition of partial differentiation, and using the rules of limits, along with a modest amount of basic algebra, I came up with this: