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Let's take a constant function for example f(x)=2 , x$\in$R. In every point of domain of f we have local minimum and maximum becouse : Let a $ \in $ R. Then exist $\alpha$ >0 that for every $x\in$B(a,$\alpha$) we have f(x) $\ge$f(a) and f(x) $\le$ f(a). Am i thinking correcly ?

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    $\begingroup$ Depends on the exact definition. How do you distinguish a local maximum from a minimum from a saddle point? I'd say "no" because $f(x) \ge f(a)$ is not a strict inequality. But your book's definition may not make that distinction. $\endgroup$ – fleablood Jan 31 at 21:21
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    $\begingroup$ Hmm.... guess I was wrong about basic definitions.... SO yes you are correct. It's my opinion concepts and theory and understanding implications are more important than precise definitions. ANd you definitely have that correct. $\endgroup$ – fleablood Jan 31 at 21:52
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Yes, you are. Note that you wrote that there is a $\alpha>0$ such that…, but one can say more here: every $\alpha>0$ will do. Of course, all that is needed is that there is some $\alpha>0$.

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Yes, that is correct. For a function $f:D\to\mathbb R$ where $D\subseteq\mathbb R$, a point $x\in D$ is a local maximum if $f(x)$ is at least as large as each point in the image of the local neighborhood of $x$; i.e. there exists an $\epsilon>0$ s.t. $f(x)\geq f(y)$ for each $|x-y|<\epsilon$. The definition is similar for a local minimum. So just choose $\epsilon=1$ and your conclusion will of course follow.

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