Suppose the function f:R^2→R has 1st order partial derivatives and that
δf(x,y)/δx = δf(x,y)/δy = 0 for all (x,y) in R^2.
Prove that f is constant; there exists c such that f(x,y) = c for all (x,y) in R.
There's a hint as well:
First show that the restriction of f:R^2→R to a line parallel to one of the coordinate axes is constant.
The chapter defines partial derivatives in general:
Let O be open in R^n and i be an index with 1≤i≤n. A function f:O→R has a partial derivative with respect to its ith component at the point x in O provided that
lim_(t→0)(f(x + t*e_i) - f(x))/t exists.
The Attempt at a Solution
The chapter has no mention of integrals but I tried using integrals anyway since I have no idea what to do with the given hint.
∫δf(x,t)/δy dt +
= f(x,y) - f(x,0) + f(x,0) - f(0,0)
= f(x,y) - f(0,0)
implying f(x,y) = f(0,0) for all (x,y).
But then I realized that the partial derivatives being 0 means their antiderivatives are constant, so that wont work out.
Now I'm looking at the hint again but, I'm really stumped on how to use it. The chapter mentions the Mean Value Theorem. I thought of trying that somehow, but it's not given that the function is continuous. Can anyone point me in the right direction?