This problem has me stumped. I'm not sure how to proceed.
Let $A = (0,\infty)$ and let $k: A \to \mathbb{R}$ be defined as follows: $ k(x) = \begin{cases} 0 & \text{for } x \in \mathbb{R}_+\setminus\mathbb{Q} \\\\ n & \text{for } x = \frac mn \in \mathbb{Q}_+ \text{ with } (m,n) = 1 \end{cases}$ Prove that $k$ is unbounded on every open interval in $A$. Conclude that $k$ is not continuous at any point of $A$.