# An interesting application of Baire's category theorem

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!

• Is $f$ continuous? – Saucy O'Path Mar 24 '19 at 15:10
• It's not stated to be continuous in the source I'm reading. – MrFranzén Mar 24 '19 at 15:10
• That makes sense. In fact, for continuous functions this would be obvious. – Saucy O'Path Mar 24 '19 at 15:11

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 $$f^{-1}((-1,1)),f^{-1}((-2,2)), f^{-1}((-3,3))\dots$$