# Terminology: Name for the value at which a piecewise function changes which piece is “active”

Consider the function $y = |x|$. It has two pieces:

1. The first is "active" for $x ≤ 0$
2. The second is "active" for $0 < x$.

$0$ is like a "tipping point" that changes which of the two pieces are active.

Is there a name for a value like $0$ in this example?

• I'd suggest critical value but why would you require such terminology? – Karl Jun 16 '17 at 19:42
• @Karl I need to come up with a name for a variable :p There are 2 hard problems in computer science: cache invalidation, naming things, and off-by-1 errors. – Alexander Jun 16 '17 at 20:08
• In this case, it is the 'cusp', or non-differentiable point. But in general it may be differentiable. Suppose you define $f(x)$ to be $x^2$ for $x <3$ and $x^2$ for $x \geq 3$. Is $3$ the 'tipping point' you want? There is nothing special about it in terms of the function, but only in the particular definition you gave. – Jair Taylor Jun 16 '17 at 21:11
• @JairTaylor Yeah it's not necessarily non-differentiable, it's just the value of x that shifts the value of $f(x)$ from one piece of the function to another – Alexander Jun 16 '17 at 21:26

I'd suggest "Boundary of the intervals of definition" or "Boundary point of the intervals on which the function $f$ was piecewise defined". If you have to use this name more than a couple times, you might consider your own name for it. I have not heard any standard name however.