TITLE: Rudin's RCA Q3.4
QUESTION [5 upvotes]: I'm trying to solve the following question from Rudin's Real & Complex Analysis. (Chapter 3, question 4) :

Suppose $f$ is a complex measurable function on $X$, $\mu$ is a positive measure on $X$, and
  $$\varphi(p) ~=~ \int_X |f|^p \; d\mu ~=~ \|f\|_p^p,~~~~~~~~~~(0 < p < \infty).$$
  Let $E := \big\{ p :~ \varphi(p) < \infty\big\}$. Assume $\|f\|_\infty > 0$.
(a) If $r < p < s$, $r \in E$, and $s \in E$, prove that $p \in E$.
(b) Prove that $\log(\varphi)$ is convex in the interior of $E$ and that $\varphi$ is continuous on $E$.
(c) By (a), $E$ is connected. Is $E$ necessarily open ? Closed ? Can $E$ consist of a single point ? Can $E$ be any connected subset of
  $(0,\infty)$ ?
(d) If $r < p < s$, prove that $\|f\|_p \leq \max\big( \|f\|_r, \|f\|_s\big)$. Show that this implies the inclusion
  $$ \mathcal{L}_r(\mu) \cap \mathcal{L}_s(\mu) ~\subseteq~\mathcal{L}_p(\mu).$$
(e) Assume that $\|f\|_r < \infty$ for some $r < \infty$ and prove that $$ \|f\|_p \xrightarrow[p \rightarrow \infty]{}\|f\|_\infty.$$

I got a solution to (a), (d) and (e). While typing my question, MSE suggested me to look at $a\mapsto \log\left(\lVert f\lVert_{1/a}\right)$ is a convex map
which seems to be related to (b). However I'm clueless about (c). Where should I start from ?
Edit
I believe the idea from $a\mapsto \log\left(\lVert f\lVert_{1/a}\right)$ is a convex map can be applied to get a proof that
$p \mapsto \log\|f\|_{\frac{1}{p}}$ is continuous on the interior of $E$. But this is not quite what we are asked to demonstrate... I'm puzzled.
Second Edit
Based on zhw's answer, it appears that $E$ is not necessarily open, nor necessarily closed and that it can be a singleton. The question whether or not $E$ can be any connected subset of $(0,\infty)$ remains. But I think I can come up with a proof that it can.

REPLY [2 votes]: This is an answer for the continuity of $\varphi.$
WLOG $f\ge 0.$ Let $A=\{f\le 1\},$ $B= \{f> 1\}.$ Suppose $[p,q]\subset E.$ Let $p_n\to p$ within $[p,q].$ Clearly $f^{p_n}\to f^p$ pointwise everywhere.
Now
$$\tag 1\int_X f^{p_n} = \int_X f^{p_n}\chi_{A} + \int_X f^{p_n}\chi_{B}.$$
Observe $f^{p_n}\chi_{A} \le f^{p}\chi_{A}$ and $f^{p_n}\chi_{B} \le f^{q}\chi_{B}.$ Since both $f^{p}\chi_{A}, f^{q}\chi_{B} \in L^1(X),$ the DCT implies the right side of $(1)$ converges to
$$\int_X f^{p}\chi_{A} + \int_X f^{p}\chi_{B} = \int_X f^{p}.$$
This implies $\varphi$ is continuous from the right at $p.$
Similarly, $\varphi$ is continuous from the left at $q.$ It follows that $\varphi$ is continuous on $E.$