Name of functions that can't be written in as a simple equation.

Question

For example a step function: $$f(x)=\begin{cases}\phantom{-}1, & \text{if }\: x>0 \\ -1, & \text{if }\: x<0 \end{cases}$$ or a function that is two functions combined: $$f(x,y)=\begin{cases}xy, & \text{if }\:x>0 \\[1ex] \dfrac{y}{2x}, & \text{if }\:x<0 \end{cases}$$ Is there a name for functions like this that require a case statement to write out?

Answer 1

Short Answer

The examples in the question are piecewise functions or piecewise-defined functions. There is not, to my knowledge, a universally accepted term which covers every possible function defined in cases, though the term casewise function would likely be understood..

Long Answer

I am not aware of a commonly used term which describes any function which is defined by different formulae on different sets. That being said, I think that the terms casewise function or casewise-defined function would get the job done—I am not aware of either term appearing in the literature anywhere, and it may very well be my own invention, but, again, I think that the terms would be understood in the intended fashion. Wikipedia also suggests that such a function could be described as "defined by cases", or may be called a hybrid function. In the comments below, bof suggests the alternative split function.

However, the examples given in the question are more specific than a function which is defined over arbitrary subdomains: they are piecewise functions, or piecewise-defined functions (see [1], [2]). A piecewise function is a function which is defined by different formulae on a sequence of disjoint intervals (including possibly singleton points, which are a kind of degenerate interval, i.e. $\{x\} = [x,x]$). Using some extra notation:

Definition: A piecewise function is a function $f : D \to \mathbb{R}$ defined by a formula of the form $$ f(x) = \begin{cases} f_1(x) & \text{if $x\in I_1$,} \\ f_2(x) & \text{if $x\in I_2$,} \\ \vdots & \vdots \\ f_N(x) & \text{if $x\in I_N$,} \end{cases} $$ where $N$ is a natural number, $\{I_n\}_{n=1}^{N}$ is a sequence of pairwise disjoint intervals, $D = \bigcup_{n=1}^{N} I_n \subseteq \mathbb{R}$, and $\{ f_n : I_n \to \mathbb{R}\}_{n=1}^{N}$ is a sequence of functions.

In short, the domain of a piecewise function is broken into a sequence of "pieces" (intervals), and the function itself is defined differently on each piece. Note that the codomain could be any space, but it is, perhaps, easier to think about real-valued functions.

There are examples of casewise-defined functions which are not piecewise-defined, at least as the term has been defined above. For example, the characteristic function of the rationals, i.e. $$ \chi_{\mathbb{Q}}(x) := \begin{cases} 1 & \text{if $x\in\mathbb{Q}$, and} \\ 0 & \text{otherwise.} \end{cases} $$ Because $\mathbb{Q}$ cannot be written as the disjoint union of a finite number of intervals, this function is not piecewise-defined, though it is certainly defined by cases.

I'll also note, for the sake of completeness, that you are unlikely to be misunderstood if you say that the characteristic function of the rationals is piecewise-defined, and I am certain that you could find authors who would use this terminology.