What Is a Continuous Partial Derivative in Two Variables?

[Shaybay92]

My textbook describes how some functions are not well approximated by tangent planes at a particular point. For example

f(x)= xy / (x^2 + y^2) for x /= 0
0 for x = 0

at (0,0) the partial derivatives exist and are zero but they are not continuous at 0. What exactly is a 'continuous partial derivative' in two variables? How do you visualize this?

 

[radou]

A partial derivative is a function, so speaking of continuity makes perfect sense.

This should help:

http://www.math.tamu.edu/~tvogel/gallery/node14.html

 

[Shaybay92]

In the case of that example, is it not differentiable at zero because its not continuous there?

 

[arildno]

In the case of that example, is it not differentiable at zero because its not continuous there?

Correct.

A function is only differentiable at zero if a unique tangent plane can be assigned there.

Differentiability IMPLIES existence of partial derivatives, but the converse does not hold.

 

[Shaybay92]

Thanks! By the way, nice job on the 9,999 posts :)