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

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The discussion centers on the mathematical proof that if the gradient of a function f: Rn -> R is zero at all points within an open ball B(a, r), then the function must be constant throughout that ball. The key argument involves demonstrating that for any two points within B, a straight line can be drawn that remains entirely within B, allowing the application of one-variable calculus principles. Specifically, if f is differentiable and f'(x) = 0 on an interval, then f is constant on that interval, which directly supports the conclusion for the multi-variable case.

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

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