Looking for a better counterexample I'm supposed to find a counterexample to counter this claim
Suppose $f: X \mapsto Y$ is a continuous function. Suppose X is bounded then $f(x)$ is bounded. My counterexample is this:
Let $X = (-\pi/2, \pi/2)$. A is bounded by $B_{\pi/2}(0)$. Let $f(x) = \tan(x)$ and let $Y = \mathbb{R}$ and $\mathbb{R}$ is unbounded.
First of all, is my counterexample right? Second, is there simpler solution?

Your counterexample is right. My go-to solution for this is $X = (0, 1]$, $Y = [1, \infty)$ and $f(x) = 1/x$.
