EDIT: Sorry, I think I posted this in the wrong section. I have moved it to "

Analysis, Topology and Differential Geometry". Please delete it!

Hi,

I have a past tier exam problem which I would like to check my solution for.

The question: Let p be real. Suppose is continuously differentiable, and satisfies

for all and for all .

Let denote the gradient of f at x and the dot product. Prove that

for all .

I first considered the n=1 case: Fixing , define 0,\infty)\to \mathbb{R}" alt="g0,\infty)\to \mathbb{R}" /> by . We have for all . Differentiating with respect to gives .

On the other hand, .I do not know how to justify this step however: f is defined on R-0, so how does one go about differentiating with respect to a variable that has a restricted domain?I would then have , and taking gives the desired result.

Onto the general case, fixing and defining again, we have and on the other hand, by the chain rule,

. Taking again gives the result.

Is the above jump to Rn-0 ok?

Thanks so much for your help!