Let us say that $f$ is a function belonging to $L^p(0,1)$. Then nothing assures us that $f$ belongs to $L^q (0,1)$ for some $q > p$. I was trying to find an "optimal function" for its $L^p$ space, in other words I want to find a function such that $f \in L^p(0,1)$, but $f \notin L^q (0,1)$ for any $q > p$.
My first attempt was to consider the following type of functions: $$ f(x) = \frac{1}{x^\frac{1- \epsilon}{p}} $$ for some small $\epsilon > 0$, anyway such class of function is not optimal in the sense given above; in fact $f \in L^{p + \delta} (0,1)$ for any $\delta > 0$ such that $$ (p+ \delta) \frac{1- \epsilon}{p} < 1 $$ Such inequality is satisfied for any $\delta < \frac{p}{1-\epsilon} - p$ and the last term is strictly greater than $0$ since $1 - \epsilon < 1$.