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Compute $f_{xy}(0,0)$ and $f_{yx}(0,0)$ and also discuss the continuity of these two at that point. given that

$$f(x,y)={{xy^3}\over {x+y^2}},(x,y) \ne (0,0)$$ and $$f(0,0)=0.$$

I was able to find the two derivatives as $f_{xy}(0,0)=0$ and $f_{yx}(0,0)=1$. How do i go about the second part of the question where it asks to discuss the continuity ? Do i need to find the two partial derivatives and they check their continuity or there is a theoretical work around ?

Grigory M
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Aman Mittal
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  • I suggest that you compute the partial derivatives. You should find http://www.wolframalpha.com/input/?i=D[x+y^3%2F%28x%2By^2%29%2Cx%2Cy] and http://www.wolframalpha.com/input/?i=D[x+y^3%2F%28x%2By^2%29%2Cy%2Cx] – Siminore May 05 '14 at 14:47
  • are you suggesting that the partial derivatives should actually be found and then checked for continuity ? – Aman Mittal May 05 '14 at 14:53
  • Yes, of course. – Siminore May 06 '14 at 09:23

1 Answers1

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Since continuity is in question, investigating the derivative would not help (since there are continuous but nondifferentiable functions or worse see http://en.wikipedia.org/wiki/Multivariable_calculus). See http://www.math.jhu.edu/mathcourses/202/Florin_notes/notes_02-15-05.pdf as an example of how to investigate the continuity of multivariable.

user72272
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