# Definition/characterisation of an almost surely constant continuous random variable

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 $$c\in\mathbb{R}$$.
• 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. – C. Brendel Jun 6 at 16:12
• 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 $P(X\leq 1)=1$) – Mark Jun 6 at 17:05