As a statistical mathematician, does this upset you?

Question

Studying statistics I stumbled upon the cumulative distribution function being defined as

https://wikimedia.org/api/rest_v1/media/math/render/svg/e5c664c7665277eea8f74575f4650fa933f28dcb

I thought to myself, why isn't it written the following way?

for $x \leq a$
for $a \leq x \leq b$
for $x \geq b$

So I asked a supporting teacher, and he wasn't 100% sure, but he jokingly said that doing this probably would upset a mathematician somewhere, thus the question in the heading. Would this upset you?

Answer 1

Your set of conditions doesn't upset me. It does make me expect there is an error in your set of conditions. Particularly, ...

The set of conditions you write is incorrect in the case $a = b$. When $a = b$ in the original set of conditions, we find the CDF is zero to the left, one to the right, and undefined at $x = a = b$. When $a = b$ in your set of conditions, we find that the CDF must simultaneously be $0$, $1$, and undefined at $x = a = b$. Consequently, the original conditions are coherent when $a = b$ and your set of conditions are not coherent when $a = b$.

Answer 2

Referring to the Wikipedia article, the parameters are usually defined: $$-\infty<a<b<\infty$$ and the interval can be closed, open or semi-open: $$[a,b] \quad \text{or} \quad (a,b) \quad \text{or} \quad [a,b)$$ Hence, the CDF can be defined accordingly: $$F(x)=\begin{cases}\ \ 0, \ \ \ \ \ \ \ \ \ \ x<a\\ \frac{x-a}{b-a},a\le x\le b\\ \ \ \ 1, \ \ b<x\end{cases} \\ F(x)=\begin{cases}\ \ 0, \ \ \ \ \ \ \ \ \ \ x\le a\\ \frac{x-a}{b-a},a< x< b\\ \ \ \ 1, \ \ b\le x\end{cases}\\ F(x)=\begin{cases}\ \ 0, \ \ \ \ \ \ \ \ \ \ x< a\\ \frac{x-a}{b-a},a\le x< b\\ \ \ \ 1, \ \ b\le x\end{cases}$$ Note that the CDF is a piecewise function, whose domain must be non-overlapping. See this.

So, probabilistically, both ways produce equal results: $x=a \Rightarrow F(a)=0$ and $x=a \Rightarrow F(a)=\frac{a-a}{b-a}=0$, however, the definition of piecewise function is violated.