Let $f\geq 1$, Is the function $p\rightarrow \int |f|^p d\mu$ continuous

Question

Let $f:X\rightarrow [0,\infty[$ be a measurable function that is greater than or equal to $1$ for every $x\in X$ and $\mu$ be a positive measure on $X$. Consider the function $g:]0,\infty[\rightarrow [0,\infty]$ that sends $p$ to $\int_X f^pd\mu$, must $f$ be continuous ?

I think the answer is yes, but I did not succeed in finding anything useful. I prefer hints rather than full answers

Thanks in Advance

Answer 1

No, $g$ need not be continuous. It can have a jump discontinuity if you have an $f$ such that $\int_X f^{p_0}\,d\mu < \infty$, but $\int_X f^p\,d\mu = \infty$ for all $p > p_0$. An example for such an occurrence is

$$f(x) = \frac{1}{x(\log x)^2}$$

on $(0,1/4)$. $f \in L^1((0,1/4))$, but $f \notin L^p((0,1/4))$ for all $p > 1$.

However, that is the only type of discontinuity that can occur, for points where $g$ is finite in a neighbourhood of $p$, the monotone convergence theorem asserts the continuity.