Differentiable function on Rn

(I moved this from the calculus section)

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 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!