Local Extremum at a Point

Let $I$ be an interval in the sense of Interval in the Real Line, let $f:I\to\mathbb{R}$, and let $c\in I$. One says that $f$ has a local extremum at $c$ if either $f$ has a local maximum at $c$ or $f$ has a local minimum at $c$; that is, there exists $\delta>0$ such that for every $x\in I$ with $|x-c|<\delta$, either $f(x)\le f(c)$ for all such $x$ or $f(x)\ge f(c)$ for all such $x$.