Flat function

In real analysis, a real function is defined to be flat at a point in its domain if all its derivatives or partial derivatives exist at that point and equal ${\displaystyle 0}$.

A real function is locally constant (that is, constant in at least one neighbourhood) of a point in the interior of its domain if and only if the function is flat and analytic at that point.

An example of a function that is flat only at an isolated point is ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$ such that ${\displaystyle f(0)=0}$ and that for all ${\displaystyle x\in \mathbb {R} }$, ${\displaystyle x eq 0}$ implies ${\displaystyle f(x)=e^{-1/x^{2}}}$; the function ${\displaystyle f}$ is flat only at ${\displaystyle 0}$.

Since ${\displaystyle f}$ is not analytic at ${\displaystyle 0}$, the extension of ${\displaystyle f}$ to ${\displaystyle \mathbb {C} }$ is not holomorphic at ${\displaystyle 0}$, since for complex functions, holomorphicity at a point implies analyticity at that point.