Hey everyone, I just want to make sure that I did this problem correctly. I have to show that this is not differentiable at the indicated point.
And this is what I did:
So at (0,0) this is undefined and it is the same for . Thus is is not differentiable at the indicated point. Is this correct? Thanks for your help!
Not quite. What you proved is that cannot be continuously extended at 0 (the limit depends on the sign of when it approaches 0).
The following works: if were differentiable at , then the function would be differentiable at (it is just saying that if is differentiable at , then exists at (0,0), since this is a "directional derivative" at ). However, as you know, this is false.
The function is continuous at 0, is not continuous at 0 (as you can check by computation). However, is differentiable at 0 ( as tends to 0)...
Your argument could be made rigorous using additional properties. You can find two directions ( for ) where different finite limits exist for . And then use the following fact: if a continuous function is differentiable on and such that its derivative admits a finite limit at , then the function is differentiable at and its derivative at is . But this is a bit overcomplicated.