I've just started working with $L^p$ spaces, and I've learned that a function $u:\mathbb{R}^n \rightarrow \mathbb{R}$ is essentially bounded if there exists a constant M such that $\{x \in \mathbb{R}^n : |u(x)| > M\}$ is of measure zero. We also write this as $u \in L^{\infty}(\mathbb{R}^n)$.
If we assume that some $u \in L^{\infty}(\mathbb{R}^n)$ is also continuous, does it imply that $u$ is bounded, not just essentially bounded?
I think it would be possible to prove this by contradiction; if we assume $u$ is unbounded on a set of measure zero, by continuity we can find a small (but of nonzero measure) region around a point with arbitrarily large $u(x)$. However, I haven't really done any measure theory so I'm not sure how to go about the details.