If $U \sim \operatorname{Unif}(0,1)$, why is $F(a) = a$ for arbitrary $a \in \mathbb{R}$

Question

Let $U \sim \operatorname{Unif}(0,1)$ be a random variable. Then the cdc $F$ of $U$ is

$$ F(x) = \left\{ \begin{array}{ll} 0 & \quad x < 0 \\ x & \quad 0 \leq x < 1\\ 1 & \quad x\geq 1 \end{array} \right. $$

The other day my professor mentioned that for all $a \in (-\infty,\infty)$, we have

$$F(a) = a$$

But I don't see how this is true. If $a < 0$, then $F(a) = 0$, and if $a \geq 1$, we have $F(a) = 1$. It's only when $0 < a < 1$ that we have $F(a) = a$.

I'm not sure if I'm missing something, because it definitely does not seem that $F(a) = a$ is true for all $a \in \mathbb{R}$.

Answer 1

You're right, it's only for $a$ in $[0,1]$. You either misheard your professor or they were wrong.