Let $f\in L^{\infty}(\Omega,\Sigma,\mu)\cap L^{1}(\Omega,\Sigma,\mu)$. Then $w(p)=||f||_p$ is continuous function of $p$ for any $p\in [1,\infty)$. How to prove this?
I have obtained the proof that $\lim\limits_{p\to\infty}{||f||_p}=||f||_\infty$. But I do not know how to prove for an arbitrary real number in $[1,\infty)$.
Your assumptions guarantee that $\| f\|_p<\infty$ for all $p\in [1,\infty)$. One may also assume without loss that $f$ is not zero a.e., for otherwise there is nothing to prove. The function $\varphi(p):=p\ln \| f\|_p$ is then convex on $[1,\infty)$, by Holder's inequality. As it is finite valued for all $p\in [1,\infty)$ it is continuous. Hence $w(p)=\exp(p^{-1}\varphi(p))$ is continuous.
