Can a Function be Constant on an Open Ball with a Zero Gradient at All Points?

[eridanus]

Let f : Rn -> R.
Suppose that grad(f(x)) = 0 for all x in some open ball B(a, r).
Show that f is constant on B(a, r).
[Hint: use part (a) to make this a problem about a function of one variable]

part (a) is show that for any two points x, y in B
there is a straight line starting at x and ending at y that is contained
in B, which I got, but I don't understand what it has to do with anything. Isn't this just a property of the gradient?

Any help would be greatly appreciated.

 

[Galileo]

Well, try the one-variable case first. Suppose you have a differentiable function f:R->R and f'(x)=0 on some open interval (a,b). Show that f is constant on (a,b).

Then, note that grad(f)(x)=0 are three equations and mix it with the result of part (a).

 

[HallsofIvy]

If the gradient is 0 at every point, then the derivative along any line is 0.