\subsection{Real-valued functions}\label{subsec:real_valued_functions}

\begin{definition}\label{def:epigraph}
  Let \( X \) be an arbitrary set. The \term{epigraph} of the function \( f: X \to \BbbR \) is defined as
  \begin{equation*}
    \epi f \coloneqq \{ (x, r) \in X \times \BbbR \colon r \geq f(x) \},
  \end{equation*}
\end{definition}
