Absolute maximum on $(-\infty, \infty)$, must be local maximum?

Question

I think yes, and I think it must true for any open interval not just $(-\infty, \infty)$

Just wanted confirmation, OR am I wrong

Answer 1

Let $M\in\mathbb{R}$ be an absolute maximum on $(-\infty,\infty)$ for a function $f$ defined in $\mathbb{R}$.

Then $\forall x\in\mathbb{R},\,f(x)\le M$.

Let $x_0\in\mathbb{R}$ and let $\varepsilon>0$.

$\forall x\in ]x_0-\varepsilon,x_0+\varepsilon[,\,x\in\mathbb{R}$ and so $\forall x\in ]x_0-\varepsilon,x_0+\varepsilon[,\,f(x)\le M$ thus $M$ is a local maximum at any point of $\mathbb{R}$.