Intuition behind Schwarz theorem (partial derivatives)

Solution 1:

The intuition is in the proof (page 49 here https://mtaylor.web.unc.edu/wp-content/uploads/sites/16915/2018/04/analmv.pdf), which proceeds by approximating $\partial_k$ with the finite difference quotient operator $\Delta_{k, h}$ defined by $\Delta_{k, h}f(x) = \frac{f(x + he_k) - f(x)}{h}$. I will write the proof here. We have \begin{align} \Delta_{k,h}\Delta_{j, h}f(x) &= \partial_k(\Delta_{j, h}f)(x + \theta_khe_k) \\ &= \Delta_{j, h}(\partial_k f)(x + \theta_khe_k) \\ &= \partial_j\partial_k f(x + \theta_khe_k + \theta_jhe_j), \end{align} where $\theta_k, \theta_j \in (0, 1)$. Thus if $\partial_j\partial_kf$ is continuous at $x$, then $$\lim_{h \to 0}\Delta_{k,h}\Delta_{j, h}f(x) = \partial_j\partial_kf(x).$$ Then note that $\Delta_{k, h}\Delta_{j, h} = \Delta_{j, h}\Delta_{k, h}$, so by swapping the roles of $j$ and $k$, it follows that if $\partial_k\partial_jf$ is also continuous at $x$, then $\partial_k\partial_jf(x) = \partial_j\partial_kf(x)$. You can extend the result to higher indices by induction, or by repeating the proof.