For a continuous random variable $X$, with probability density function $p_X(x)$, it is known that there exists a $p_{min} > 0$ such that $p_X(x) \geq p_{min} \forall x \in X$. Also, I know that $X$ is bounded : $\exists M > 0 : X \subseteq [-M, M]$.
Let $r > 0$ be fixed (this is a small number). Define $S_r = \{ x\in [-M, M] : p_X(x) > r^{-\alpha} \}$, for some $\alpha > 0$. Also, let $b\in X$. Then I want to find an upper bound for:
$$\mathbb{P}(X \in S_r \cap [b-r, b +r] )$$
This upperbound will need to be small when $r$ is small. Notice that $r$ is fixed, so I am not looking for an upperbound that is $0$ in the limit, but a stronger statement.