---- > [!definition] Definition. ([[characteristic function]]) > Let $X$ be a set and $E \subset X$. The **characteristic function of $E$** is the function $x \mapsto \begin{cases} 1 & x \in E \\ 0 & x \not \in E. \end{cases}$ It is variously denoted $\chi_{E}$, $1_{E}$, $\mathbb{1}_{E}$, etc. ^definition > [!basicexample] > The **unit step function** or **Heaviside step function** on $A \subset \mathbb{R}$ (usually $A=\mathbb{Z}$ or $A=\mathbb{R}$) is $H:=\chi_{\{ x \in A : x \geq 0 \}}.$ ^basic-example ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```