If $\nabla f(x) = 0$ on a set $S$, is $f$ necessarily constant on $S$?

Question

If $\nabla f(x) = 0$ on a set $S$, is $f$ necessarily constant on $S$?

I know it is true when $S$ is open convex, or open connected, but what about any arbitrary $S$?

Answer 1

if $S$ is disconnected, give $f$ a different constant value on each component of $S$ and we see that the conjecture is false.

Answer 2

If you define $S$ to be the disconnected set above and the height of the line above a horizontal line to be the value of $f(x)$, you can see how it is false.

This can be extended to higher dimensions in an analogous way.