Let $V$ be a real [vector space](/page/Vector%20Space) and let $C \subset V$. Define the extended-valued indicator function
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\begin{align*}
\iota_C: V &\to \mathbb{R} \cup \{+\infty\}
\end{align*}
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by setting $\iota_C(x)=0$ for $x\in C$ and $\iota_C(x)=+\infty$ for $x\notin C$. Then $\iota_C$ is convex in the extended-valued sense if and only if $C$ is convex.