Let $a < b$ and $]a,b[$ be an open real interval and $f:]a,b[\to\mathbb R$ a function $f$ is said to have a local maximum (respectively a local minimum) $x$, if there is a positive number $\epsilon > 0$ such that
$f(x)\ge f(\xi)$ (respectively $f(x)\le f(\xi)$) for all $\xi$ with $|x-\xi| < \epsilon.$
If the equality $f(x)=f(\xi)$ holds only for $\xi=x$, then the local maximum (respectively minimum) is said to be isolated.
The general term for a local maximum (or a local minimum) is local extremum (plurals: extrema, maxima, mimima).
