关于区域指示型分段函数映射的命名规则及通用名称咨询

Hey Bruno! Great question—let's break this down clearly for you.

The function you've defined here is commonly called the Indicator Function (sometimes also referred to as the Set Characteristic Function, but quick heads-up: don't mix this up with the characteristic function used in probability theory, which is a completely different concept).

Typically, it's written using shorthand notation like $\mathbf{1}_R(x)$ or $\chi_R(x)$, where:

• $\mathbf{1}_R(x) = 1$ if $x \in R$
• $\mathbf{1}_R(x) = 0$ if $x \notin R$

This matches exactly the piecewise definition you provided:

$$
y = \left{
\begin{array}{cl}
1, & \mbox{for abcisse } x \in R \
0, & \mbox{for abcisse } x \in X-R
\end{array}
\right.
$$

To address your note about the delta function: You're right that $\delta(x)$ (the Dirac delta function) is concentrated solely at the single point $x=0$ (it's a distribution, not a regular function, which is why we talk about its "measure" behavior). The indicator function is the opposite in a way—it takes the value 1 across an entire set $R$, rather than being localized to a single point.

备注：内容来源于stack exchange，提问作者Bruno Peixoto