Differentiable function on Rn
(I moved this from the calculus section)
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!