00:01
In this question, we are asked to show rigorously that this function, x times y squared divided by x squared plus y to the fourth is not differentiable, even though the partial derivatives at zero do exist.
00:18
So we're trying to show that it's not differentiable at zero.
00:23
So to see that the partial derivatives do exist is actually surprisingly easy in this case, because there are so many zeros floating around.
00:33
So if we look at the formal definition of the partial derivative with respect to x, which is the top line, plugging it in, we get this complicated looking limit, but it turns out everything inside the limit is just zero.
00:49
It's zero divided by h, which is plain old zero.
00:53
And the same goes for the fdy.
00:56
So these are actually, this is very easy to see that once you plug it, in every zero into the function, these partial derivatives exist and are zero.
01:09
So you might note, wasn't there a theorem that said if the partial derivatives exist, it's differentiable.
01:17
Almost.
01:17
So the partial derivatives need to exist in some tiny little area around the point you're looking at.
01:24
And they need to be continuous.
01:26
And that's not going to be the case here.
01:29
So that's one way you could check it, but finding the partial derivative.
01:33
Derivatives at a point at not zero is actually quite annoying because well in this case the limits are easy because everything is zero but if we're not at the point zero that doesn't go and these limits can become quite hard so what are we going to do is we're going to use another theorem in my edition of the book this is called theorem four and it says that if f is differentiable it certainly is is continuous.
02:10
Continuous.
02:13
Ok, so let's check that this function isn't continuous.
02:17
Now remember, continuous means that whichever path we take to a point, the function, the limit of this path will not change.
02:29
So more concretely, if we draw an x -y plane, so we have x and we have y, there are multiple ways in which we can walk to zero.
02:42
For example, we can just walk along the x -axis, or you can walk along the y -axis, or you can do something funny and walk like this.
02:56
And if the function is continuous at zero, all of these paths will give you the same answer...