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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.

jjagmath
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koorkevani
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    For the partial derivatives, LaTeX (and MathJax) have a special delta \partial ($\partial$) to use instead of \delta ($\delta$). – jjagmath Apr 06 '23 at 11:50
  • In the title you write “existence of $f_{yx}$” but in the question there's just $\partial f/\partial y$. Which one should it be? – Hans Lundmark Apr 06 '23 at 12:02
  • @HansLundmark My question is about both the stronger claim and the weaker claim. It starts only with $f_y$, but then says "If this is false, does requiring that $f_{yx}$ exist at $A$ make it true?" – koorkevani Apr 06 '23 at 12:12
  • OK, I understand. – Hans Lundmark Apr 06 '23 at 12:16
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    This could be of interest: https://math.stackexchange.com/questions/3291919/assumptions-for-a-variation-of-clairauts-theorem, https://math.stackexchange.com/questions/98514/existence-of-mixed-partials-in-clairauts-theorem – Hans Lundmark Apr 06 '23 at 12:32

1 Answers1

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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 $\partial_{xy} f = \partial_{yx} f$.

cs89
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    The result I stated includes the assumption that $\partial_y f$ exists on a neighborhood of $A$ (not only at $A$), which is not stated in your question, but it seems reasonable to assume this if you are going to differentiate $\partial_y f$. – cs89 Apr 06 '23 at 12:49