Timeline for Does $f$ have a critical point if $f(x, y) \to +\infty$ on all horizontal lines and $f(x, y) \to -\infty$ on all vertical lines?
Current License: CC BY-SA 4.0
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| May 25, 2020 at 18:05 | comment | added | zhw. | In fact, no $C^\infty$ function of the form $f(x,y)=g(x)+h(y)$ can be a counterexample, since $g'(x_0) = 0$ for some $x_0$ and $h'(y_0) = 0$ for some $y_0,$ giving $(x_0,y_0)$ as a critical point. | |
| May 21, 2020 at 14:14 | history | edited | DOmonoXYLEDyL | CC BY-SA 4.0 |
added 121 characters in body
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| May 21, 2020 at 14:11 | comment | added | DOmonoXYLEDyL | @K.defaoite Yes, you're right.. I totally forgot the $C^\infty$ condition. Adding that information may be an harder challenge. | |
| May 21, 2020 at 12:30 | comment | added | K.defaoite | This is a good counterexample. Not too sure if it meets the $C^{\infty}$ condition though, as it's not differentiable at $y=0$. | |
| May 21, 2020 at 11:16 | history | answered | DOmonoXYLEDyL | CC BY-SA 4.0 |