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Here is an interesting application of Baire's category theorem.

Let $f: \ \mathbb R \to \mathbb R $, then there exists a nonempty interval $(a,b) $ and a positive number $c $ such that for any $x \in (a,b) $ there is a sequence $\{x_n \} $ such that $x_n \to x $ and $|f(x_n)| \le c $.

I'm not able to solve it. Any help would be appreciated!

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  • $\begingroup$ Is $f$ continuous? $\endgroup$ – Saucy O'Path Mar 24 '19 at 15:10
  • $\begingroup$ It's not stated to be continuous in the source I'm reading. $\endgroup$ – MrFranzén Mar 24 '19 at 15:10
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    $\begingroup$ That makes sense. In fact, for continuous functions this would be obvious. $\endgroup$ – Saucy O'Path Mar 24 '19 at 15:11
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We might consider it as follows: given $f$, we are trying to prove there exists $c > 0 $ such that the nonempty open interval $(-c,c)$ has the property that $f^{-1}{((-c,c))}$ is dense on some nonempty open interval $(a,b)$. Suppose not; i.e. for every $c$ greater than zero, $f^{-1}(c)$ is nowhere dense. Then take the union of $f^{-1}((-1,1)),f^{-1}((-2,2)), f^{-1}((-3,3))\dots$

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