Is $p\mapsto \|f\|_p$ continuous?
It suffices to show $g(p) = \int_X |f|^p$ is continuous, because then $||f||_p = g(p)^{1 \over p} = e^{\log(g(p)) \over p}$ is continuous.
Note that $g(p) = \int_{|f| > 1} |f|^p + \int_{|f| \leq 1}|f|^p$. To show continuity of each term at some $p_0$, you can use the dominated convergence theorem; for some $\epsilon > 0$, $|f|^{p_0 + \epsilon}$ will serve as a dominating function for the first integral, and $|f|^{p_0 - \epsilon}$ will serve as a dominating function for the second (unless $p_0 = 1$ in which case just use $|f|$).