What does it mean if a function has well-dened partial derivatives everywhere in R2?

Can the partial derivatives have a place where they are not defined?

eg.

f(x) = xy / (x^2 + y^2)

is df/dx and df/dy "well defined"?

Thanks

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- Dec 19th 2008, 09:32 PMswramanwell defined partial derivatives everywhere in R2
What does it mean if a function has well-dened partial derivatives everywhere in R2?

Can the partial derivatives have a place where they are not defined?

eg.

f(x) = xy / (x^2 + y^2)

is df/dx and df/dy "well defined"?

Thanks - Dec 20th 2008, 04:05 AMmr fantastic
Reading this might help: http://www.econ.ubc.ca/sart/320lec9.pdf

- Dec 20th 2008, 04:09 AMMush
Both first order differentiations of your function lead to the denominator $\displaystyle (x^2+y^2)^2$, as the result of the quotient rule of differentiation.

For what values of x and y does a function $\displaystyle \frac{g(x)}{(x^2+y^2)^2} $ become undefined?

Well... for all positive and negative values of x and y, the denominator is finite and positive. However... what happens for x = y = 0 ?