Existence of mixed partials in Clairaut's theorem.

Question

In almost all statements of Clairaut's theorem, the equality of the mixed partials is given only after the assumption that both mixed partials exist and are continuous. In this blog post here, the author proves a stronger statement

"At some point $\vec{x}=(x_0,y_0)$, if the partials $f_x$ and $f_y$ are both continuous and if $f_{xy}$ exists and is continuous, then $f_{yx}$ exists and we have $f_{yx} = f_{xy}$."

The proof goes as follows:

Using the mean value theorem, the author obtains the statement $$\lim_{h\rightarrow 0}\ \lim_{k\rightarrow 0}\ f_{xy}(x_0 + \overline{h}, y_0 +\overline{k}) = f_{yx}(x_0, y_0)$$

for some point $\overline{h} \in (0, h)$ and $\overline{k} \in (0, k)$. Since $\overline{h}$ depends on $k$, the limit cannot be taken trivially. From the continuity of $f_{xy}$ we get

$$\mid f_{xy}(x,y) - f_{xy}(x_0, y_0)\mid < \frac{\epsilon}{2}$$

for $(x,y)$ within $\delta$ of $(x_0, y_0)$. If we take $\mid h \mid$ and $\mid k\mid$ small enough, say less than $\frac{\delta}{2}$, it follows that the point $(x_0 + \overline{h}, y_0 +\overline{k})$ satisfies the limit. So we fix $h$ small enough to obtain

$$\mid f_{xy}(x_0 + \overline{h},y_0 + \overline{k}) - f_{xy}(x_0, y_0)\mid < \frac{\epsilon}{2}$$

for $\mid k\mid < \frac{\delta}{2}$.

What he does next is totally lost to me.

He says we can take the limit from $k\rightarrow 0$, and when we do so, the inequality may become an equality (why?). But since we chose $\frac{\epsilon}{2}$ initially, we obtain $$\mid \lim_{k\rightarrow 0}\left[f_{xy}(x_0 + \overline{h},y_0 + \overline{k})\right] - f_{xy}(x_0, y_0)\mid \le \frac{\epsilon}{2} < \epsilon$$

from which the limit is established.

My main questions concern what exactly he did in the last few steps. How could the inequality possibly become an equality? And it seems to me that what he did is no different than just trivially taking $k$ to zero and then $h$ to zero.

Answer 1

The fact under consideration is true. It amounts to Fubini's Theorem, in this shape: for some continuous $h(x,y)$ define $$ G(x,y) = \int_a^x \int_c^y \; h(u,v) \; dv \; du $$ Fubini says that we can interchange the order of integration. That and the fundamental theorem of calculus say that both $$ \frac{\partial}{\partial y} \left( \frac{\partial G}{\partial x} \right) $$ and $$ \frac{\partial}{\partial x} \left( \frac{\partial G}{\partial y} \right) $$ exist, are continuous and equal to $h.$

In particular, if $ \frac{\partial}{\partial y} \left( \frac{\partial f}{\partial x} \right) $ exists and is continuous, iterated integration tells us that $$ G(x,y) = f(x,y) - f(x,c) - f(a,y) + f(a,c), $$ with $$ \frac{\partial}{\partial y} \left( \frac{\partial G}{\partial x} \right) = \frac{\partial}{\partial y} \left( \frac{\partial f}{\partial x} \right) $$ However, the other order exists for $G,$ so we get existence for $f$ and $$ \frac{\partial}{\partial x} \left( \frac{\partial f}{\partial y} \right) = \frac{\partial}{\partial x} \left( \frac{\partial G}{\partial y} \right) $$

This argument is from AKSOY_MARTELLI. I was able to view the pdf on my computer screen but not print it out. We are talking about the first part of Theorem 3, (i) implies (ii), bottom of page 128 to the first paragraph on page 129.

Answer 2

The theorem (and proof) are basically identical to Theorem 6.20 in the first edition of Apostol's Mathematical Analysis. In the second edition Apostol has replaced this result by the standard ones that assume the existence of both mixed partials, but he notes on page 360 that

if $D_r \mathbf{f}$, $D_k \mathbf{f}$ and $D_{k,r} \mathbf{f}$ are continuous in an $n$-ball $B(\mathbf{c})$, then $D_{r,k} \mathbf{f}(\mathbf{c})$ exists and equals $D_{k,r} \mathbf{f}(\mathbf{c})$.

So I think the proof is correct.

As for the last points, in evaluating the nested limit, he first uses continuity of the Euclidean norm to move the inner limit inside. The strict inequality might fail in the limit, compare e.g. the sequence $\{ 1/n \}$, where $1/n > 0$ for all elements, but the limit as $n \to \infty$ is zero.

We clearly use continuity of the one mixed partial at the point of interest, but I fail to see where we use continuity in a neighbourhood around the point. Also, I don't see where we use continuity of the first partials. I might just ask a question about that myself.

EDIT: I have asked such a question here