I came up with this when trying to prove embeddings of $L^p(0,1)$ spaces. Suppose $U =(0,1)$ (or more generally a bounded subset in $\Re^n$). Is the mapping $\phi = p \mapsto \|f\|_{L^p(U)}, \phi: [1,q] \rightarrow \Re$ continuous? Any ideas or counter examples?
Edit: This is not a duplicate with Is $p\mapsto \|f\|_p$ continuous? because of different assumption on $f$!