Is following property of absolute moments true?
If $ E[|X|^k]<\infty$ for all $k \in [a,b]$ then $f(k)=E[|X|^k]$ is continuous on $[a,b]$.
I think it should be true. It doesn't seem possible that for $f(k)$ is discontinuous.
Note that if $X$ is bounded and $k>0$ then the result follows by the dominated convergence theorem since \begin{align} |X|^{k_n} \le A^{k_n} \le B \end{align} the last inequality follows since $k_n$ is convergence sequence it must be bounded.
Therefore, \begin{align} \lim_{k_n \to k_0} E[|X|^{k_n}]= E[ \lim_{k_n \to k_0} |X|^{k_n}]= E[ |X|^{k_0}] \end{align}
I wanted to also say that I very much interested in cases when $k$ is negative.