Let $(\Omega, F, \mathbb{P})$ be a probability space, $X$ is a random variable s.t. $X[\Omega] = [-a, a), a > 0$ and that $X$ has a density function $f_X$. If we know that $\mathbb{P}_X(c) = 0$ for a fixed $c \in [-a, a)$ does it then follow that $f_X(c) = 0$?
I am aware of the fact that the probability of a continuous random variable being equal to any one point is zero. However, in a paper I am currently reading, it was proven quite extensively why $\mathbb{P}_X(c) = 0$ when $a = \pi, c = -\pi$, $X$ is a random variable in the unit circle and we are assuming that the variance is minimized at $\mu = 0$.
The paper in question and the specific portion are Intrinsic Means on the Circle: Uniqueness, Locus and Asymptotics by Hotz and Huckerman, pp. 4 and 6; theorem 3.1 i.) and proposition 3.4.
The authors claim that the density of such a random variable $X$ in a certain setting is zero at $-\pi$, but IMO their paper doesn't explicitly argue why this is the case. Therefore, I'm interested in hearing whether $f_X(-\pi) = 0$ due to the setting of the authors, or because some known measure theoretic reason.