Continuity of $f_{xy}$ and existence of $f_{yx}$ implies continuity of $f_{yx}$? Partial derivatives

Question

Let $f(x,y): \mathbb{R}^2 \rightarrow \mathbb{R} $. Is the following statement true? If:

1. $\frac {\partial^2 f} {\partial y \partial x}$ is continuous at a point $A$ and
2. $\frac {\partial f} {\partial y}$ exists at $A$

Then $\frac {\partial^2 f} {\partial x \partial y}$ is continuous at $A$.

If this is false, does requiring that $\frac {\partial^2 f} {\partial x \partial y}$ must exist at $A$ make it true?

I got into an argument with my college calculus teacher - it feels completely false to me, but I can't find a counterexample.

Answer 1

Yes, this result is true.

It is for example stated as Theorem 9.41 (page 235-236) in the book

Walter Rudin, Principles of Mathematical Analysis (1976), International Series in Pure & Applied Mathematics, McGraw-Hill, ISBN 0-07-054235-X.

Theorem 9.41: If $\partial_x f$, $\partial_{yx} f$ and $\partial_y f$ exist on some neighborhood of $A$ and $\partial_{yx} f$ is continuous at $A$, then $\partial_{xy} f$ exists at $A$, is continuous at $A$ and $\partial_{xy} f = \partial_{yx} f$.