What are the ways of characterising that a continuous random variable $X$ is constant?

For example, if $Y$ is a discrete random variable with pdf $p_Y$ then we can say $Y$ is constant almost surely if $p_Y(c) = 1$ for some $c$.

For a continuous random variable with cdf $F_X$, we could say that $F_X$ is constant if there exists $c$ such that $F_X(c) = 1$ and for all $a < c$ we have $F_X(a) = 0$. However, this characterisation seems clumsy. I'm wondering which characterisations are more straight-forward to verify, or could make a clearer definition.

For instance, here's a characterisation that I think is correct and is arguably more elegant than the previous one:

$X$ is almost surely constant if $F_X(\mathbb R) = \{0,1\}$.

I'd intuitively like to say that $P(X = c) = 1 \implies X$ is constant, but I'm not sure I can safely consider the event $\{X = c\}$ for a continuous r.v. $X$.


A continuous random variable can't be almost surely constant. If $X$ is continuous then $P(X=c)=0$ for all $c\in\mathbb{R}$.

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  • $\begingroup$ What about $X:[0;1]\longrightarrow \mathbb R, x \longmapsto 1$ with probability space $([0;1],\mathfrak B \cap [0;1],\lambda)$ and measure space $(\mathbb R, \mathfrak B)$? Then $X$ is a continuous random variable that is constant and thus also almost surely constant. $\endgroup$ – C. Brendel Jun 6 at 16:12
  • $\begingroup$ This random variable is not continuous. It doesn't have a continuous CDF, there is a discontinuity at the point $t=1$. (because $P(X\leq t)=0$ for all $t<1$ but $P(X\leq 1)=1$) $\endgroup$ – Mark Jun 6 at 17:05
  • $\begingroup$ Oh ok, I was thinking of topological continuity. Good to know there is a difference! $\endgroup$ – C. Brendel Jun 6 at 18:04
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    $\begingroup$ Yes, these are different terms. Actually, in most probability spaces we don't even have a topology. $\endgroup$ – Mark Jun 6 at 18:07
  • $\begingroup$ ah interesting, so via the cdf you are basicially pulling back the continuous property $\endgroup$ – C. Brendel Jun 6 at 18:26

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