If $\int_{0}^{\infty} |f(x)| dx < \infty$, then $f$ need not go to $0$ as $x\to \infty$, even if $f$ is continuous.

Question

I found this problem in Patrick Billingsley's book Probability and measure and I've been stuck on trying to find a counter example. If $f$ is not required to be continuous, then I think I can find a counter example (some integrable $f$ that is unbounded on a set of measure $0$). However, the textbook states that this holds even for continuous $f$, so I was curious as to what such an example would even look like.