What type of functions fail to pass the Clairaut's theorem, are there trigonometric type of functions that meet this criteria?
2026-04-06 11:12:43.1775473963
When does the Clairaut's theorem fail?
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The simplest example that I know of is $$ f(x,y) = \begin{cases} \dfrac{x y (x^2 - y^2)}{x^2 + y^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases} $$ Using the limit definitions, you can show that $f$ is twice-differentiable, but $f_{yx}(0,0) = 1$ and $f_{xy}(0,0) = -1$.