How does one prove that if function's partial derivative respect to every variable is zero, function is constant?

multivariable-calculus partial-derivative

How does one prove that if function's partial derivative respect to every variable that the function defines over is zero function is constant function? I just noticed it, but I cannot prove it.


Suppose the function $f(x,y)$ is non-constant and its partial derivatives exist everywhere. Then we have some $(x,y)$ and $(x',y')$ such that $f(x,y)\ne f(x',y')$. If $f(x,y)=f(x',y)=f(x,y')$ then these are distinct from $f(x',y')$, otherwise one of $f(x',y)$ or $f(x,y')$ is not $f(x,y)$. This means we only have to worry about one of the two variables! Lets say $f(x,y)\ne f(x',y)$ (the other cases are identical). By the Mean Value Theorem, we have some $c\in (x,x')$ such that $$\frac{\partial f}{\partial x}(c,y)=\frac{f(x',y)-f(x,y)}{x'-x}\ne 0$$ thus the partial derivatives are not everywhere $0$.

You should be able to extend this to arbitrarily many variables on your own.

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