- #1

mahler1

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Let ##f:ℝ→ℝ## be an open and continuous function. Prove that f doesn't have local extrema

The attempt at a solution.

I suppose there is some ##x_0 \in ℝ## and some ##ε>0## such that ##f(x_0)≤f(x)## for all ##x \in (x_0-ε,x_0+ε)## (the proof for relative maximum is analogue to this one). Now, I consider the open interval ##(x_0-ε,x_0+ε)##. Then, ##f( {(x_0-ε,x_0+ε)} )## is open by our hypothesis. Then, there exists ##δ>0## such that ##B(f(x_0),δ) \subset f({(x_0-ε,x_0+ε)})##. I want to come to an absurd, this means, find some ##z \in (x_0-ε,x_0+ε)## such that ##f(z)<f(x_0)##, but I don't know how to get here, I must use continuity at some point. How could I get the wanted z?

By the way, sorry for the notation ##f({(x_0-ε,x_0+ε)})##, it should be f({(x_0-ε,x_0+ε)}), with brackets, but I couldn't put it this way with latex.