Set of solutions to probability quantile equation

Question

Let $X$ be a random variable taking its values over some set $S$, with cumulative distribution function (cdf) $F$. Let $\epsilon\geq0$. We define the set $S^*(\epsilon)$ as follows: $$S^*(\epsilon)\triangleq\{x \in S:F(x-\epsilon)+F(x+\epsilon)=1\}$$

What can be said in general about the behavior of the set $S^*(\epsilon)$ as a function of $\epsilon$?

For example, $S^*(0)$ is a singleton containing only the median. But what about the case $\epsilon>0$? Is the set also a singleton? Let us consider three possible assumptions:

1. Assumption 1: $X$ is a discrete random variable with support $\mathbb N$, e.g. Poisson distribution, Binomial distribution.
2. Assumption 2: $X$ is a continuous random variable with support $\mathbb R$, e.g. Normal distribution.
3. Assumption 3: $X$ is a continuous random variable with support $\mathbb R^-$ or $\mathbb R^+$, e.g. exponential distribution.

This is a follow-up question on this.

Answer 1

Let $F$ be absolutely continuous and symmetric around $x_0$, with density $f$. Then, for every $\delta$, $f(x_0-\delta)=f(x_0+\delta)$. Integrate both sides to get $$1-F(x_0-\epsilon)=\int_{-\infty}^\epsilon f(x_0-\delta)d\delta=\int_{-\infty}^\epsilon f(x_0+\delta)d\delta=F(x_0+\epsilon)$$ Thus, if the distribution is symmetric around $x_0$, for any $\epsilon\geq 0$, $x_0\in S^*(\epsilon)$.