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
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
Reading this might help: http://www.econ.ubc.ca/sart/320lec9.pdf
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 ?