Suppose that $\|f\|_p < \infty$ for all $1\leq p < p'$, I want to know if the the following is true and in that case how to show it
$p \mapsto \|f\|_p$ is continuous on $[1,p')$
Or maybe we need to impose some more constraints such as finite measure space. In case of finite measure space, I tried to use Egoroff's theorem.