Hello everyone, I have a question that I haven’t been able to find a proof for. Hope you guys could help me with it.
Let f: R2—>R be differentiable at all points in R2 except for (0,0). It is know that it is continuous at (0,0) and that the following goes for it-
the limit of both partial derivatives (with respect to x and y) approach to 0 at (0,0), i.e.
lim(x,y)—>(0,0) (f’x(x,y))=lim(x,y)—>(0,0) (f’y(x,y))=0
I need to prove that f is differentiable at the origin.
I have been trying to prove that the partial derivatives are continuous at the origin but I was able to do so, it feels like something is missing. Am I missing something?
Let f: R2—>R be differentiable at all points in R2 except for (0,0). It is know that it is continuous at (0,0) and that the following goes for it-
the limit of both partial derivatives (with respect to x and y) approach to 0 at (0,0), i.e.
lim(x,y)—>(0,0) (f’x(x,y))=lim(x,y)—>(0,0) (f’y(x,y))=0
I need to prove that f is differentiable at the origin.
I have been trying to prove that the partial derivatives are continuous at the origin but I was able to do so, it feels like something is missing. Am I missing something?