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We show that, for every $1 \leq p < +\infty$ and for every Borel probabilitymeasure $\mathbb{P}$ over $\mathbb{R}$, every element of $L^{p}(\mathbb{R},\mathscr{B}_{\mathbb{R}}, \mathbb{P})$ is the $L^{p}$-limit of some sequence ofbounded random variables that are Lebesgue-almost everywhere differentiablewith derivatives having norm greater than any pre-specified real number atevery point of differentiability. In general, this result provides, in somedirection, a finer description of an $L^{p}$-approximation for $L^{p}$functions on $\mathbb{R}$.

Sensitive Random Variables are Dense in Every $L^{p}(\mathbb{R}, \mathscr{B}_{\mathbb{R}}, \mathbb{P})$