Suppose I have a continuous function $f$ on $K \times \mathbb{R}$ where $K$ is a compact subset of $\mathbb{R}^n$, such that $\sup_{y \in \mathbb{R}} f(x,y)$ is finite for all $x$ (so component wise bounded). Does it follow that $f$ is also bounded on $K \times \mathbb{R}$ ?
Is the $\sup_{y \in \mathbb{R}} f(x,y)$ function continuous over $K$?