A proof of the Schwarz theorem on mixed partial derivatives via elementary approximation theory (Q827548)

The author gives a proof of the following theorem due to Schwarz. Assume that \(f(x,y)\) has partial derivatives \(f_x\) and \(f_y\), and mixed derivatives \(f_{xy}\) and \(f_{yx}\) in a neighbourhood \(\Omega\) of a point \((x_0,y_0)\) and that \(f_{xy}\) and \(f_{yx}\) are continuous at \((x_0,y_0)\). Then \(f_{xy}(x_0,y_0)=f_{yx}(x_0,y_0)\). The author divides the proof into two steps. One step, which is a lemma, assumes that \(f\) and \(g\) satisfy \[ \lim_{(x,y)\to (x_0,y_0)} \frac{f(x,y)-g(x,y)}{(x-x_0)(y-y_0)}=0 \] and have partial derivatives and mixed partial derivatives of second order. The conclusion is then that the Schwarz theorem holds. The second step shows that the lemma can be applied for appropriate functions. For this second step, the proof cleverly plays with Taylor approximations and the mean value theorem. As inspiration in the background serves the situation when \[ f(x,y)=\sum_{i=1}^n u_i(x)v_i(y) \] for some differentiable functions \(u_i\) and \(v_i\).