1. Let f : R^2 ? R be the function given by f(0,0) = 0 and...

1. Let f : R^2 ? R be the function given by f(0,0) = 0 and for all other (x,y) f(x,y) = (x^2 - y^2) / (x^2 + y^2). Show that the partial derivatives of f are everywhere defined but that f is not differentiable at the origin.

Transcript

00:01 So we are given that f be the function where r square tan into r be the function given by f of 0 comma 0 is equal to 0 for all other values of x y.

00:21 F of x y is equal to x square minus y square divided by x square plus y square.

00:30 So we have to show that the partial derivative of f are everywhere defined but f is not differentiable at origin.

00:40 So let's get started to solution.

00:43 So first of all what is our claim here? that means what is we have to prove in this calculus problem that f is not differentiable at origin.

00:58 Right so it is enough to show f is not continuous at origin so let us consider the path y is equals to m into x which coming from the equation of line y is equals to mx plus c so as we are talking about origin so the intercept value is going to be 0 here that is why y is equals to mx here now we just apply the limit x, y both will tends to 0, 0 of the function f of x, y equals to the limit we say x, y tends to 0, 0.

01:39 Whatever the function is there, just put it right here.

01:43 That is x square minus of y square divided by x square plus of y square.

01:49 We can substitute the value of y here.

01:52 We will get the limit x ,y both stands to 0 will become x square minus of m square into x square divided by x square plus m square x square.

02:07 So now we can take x as common here.

02:10 This will become limit extending to 0 only because y is already eliminated.

02:16 While we taking x square as common 1 minus of m square divided by the x square common 1 plus m square...

Question

1. Let f : R^2 ? R be the function given by f(0,0) = 0 and for all other (x,y) f(x,y) = (x^2 - y^2) / (x^2 + y^2). Show that the partial derivatives of f are everywhere defined but that f is not differentiable at the origin.

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