I've seen Baire category theorem used to prove existence of objects with certain properties. But it seems there is another class of interesting applications of Baire category theorem that I have yet to see.

First I found this MathOverflow problem:

Let $f$ be an infinitely differentiable function on $[0,1]$ and suppose that for each $x \in [0,1]$ there is an integer $n \in \mathbb{N}$ such that $f^{(n)}(x)=0$. Then $f$ coincide on $[0,1]$ with some polynomial.

I found another one from Ben Green's notes:

Suppose that $f:\mathbb{R}^+\to\mathbb{R}^+$ is a continuous function with the following property: for all $x\in\mathbb{R}^+$, the sequence $f(x),f(2x),f(3x),\ldots$ tends to $0$. Prove that $\lim_{t\to\infty}f(t)=0$.

Are there any other classic problems of this type?