I'm working through Theorem 9.5 in Rudin's Real and Complex Analysis and I'm having trouble understanding one of the steps. The statement of the proof is as follows:
Rudin uses Theorem 3.14, which states that $C_C(X)$ is dense in $L^p(X)$, to obtain the function $g$ and this is where my confusion comes in. I don't understand how Rudin deduces the uniform continuity of $g$ rather than just continuity. My understanding is that $C_C(X)$ includes all continuous complex functions with compact support and not just uniformly continuous functions. Any help understanding this would be appreciated.