The standard definition of is $f(x)=\begin{cases}x,& x\geq 0\\-x,&x<0\end{cases}$. I am wondering what the problem is with the definition $f(x)=\begin{cases}x,& x,\geq 0\\-x,&x\leq0.\end{cases}$. Similarly, for piecewise defined functions like $f(x)= \begin{cases} x-1, & x\leq 2 \\ -3+4x-x^2, & x> 2 \end{cases}$ I remember having seen authors using the definition $f(x)= \begin{cases} x-1, & x\leq 2 \\ -3+4x-x^2, & x\geq 2 \end{cases}$ (see for instance Wiki's lemma for piecewise functions: https://en.wikipedia.org/wiki/Piecewise). Is there a difference between the two? Is the former preferable?
2026-03-27 05:16:00.1774588560
Absolute value function definition
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When defining a function piecewise like this, it is a simple courtesy to your readers to make sure that all your cases are mutually exclusive. That way, they don't have to check that the function takes the same value on the intersection of two or more different cases, which is required for your definition to be valid.