If $f\in L^1(\Bbb R)$ is continuous then $f\in C_0(\Bbb R)$ Suppose $f \in L^1(\Bbb R)$ is continuous. Then is it necessarily true that $f\in C_0(\Bbb R)$? It is easily seen to be true if $f$ is uniformly continuous, but I can't see whether it is true if $f$ is merely continuous. Thanks in advance.

You can even have continuous unbounded real maps that are Lebesgue integrable. See here for an example.
