Hi I would just like for someone to check my proof/point out anything I am missing. Thanks!

Prove that the function is not diff at origin.

Proof:

If it was differentiable at the origin, then should have the same derivative no matter how you approach the origin. We consider two paths approaching the origin from the x axis (positive side) and the line from the 1st quadrant.

Case 1 (x-axis):

Case 2 (x=y line):

Since these two limits are different. The function is not differentiable at the origin.