---
date: '2024-12-10'
description: supergraph of a function consisting of all points in cartesian product lying on or above the function's graph.
id: epigraph
modified: 2026-06-05 15:08:21 GMT-04:00
tags:
  - math
title: epigraph
created: '2024-12-10'
published: '2024-12-10'
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slug: thoughts/epigraph
permalink: https://aarnphm.xyz/thoughts/epigraph.md
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full: https://aarnphm.xyz/llms-full.txt
---
> \[!definition\] Definition 1.
>
> A epigraph or _supergraph_ of a function $f: X \to [-\infty, \infty]$ valued in the extended real numbers $[-\infty, \infty]=\mathbb{R} \cup \{\pm \infty\}$ is the following set:
>
> $$
> \text{epi} f = \{(x,r) \in X \times \mathbb{R} : r \ge f(x)\}
> $$

which consists of all points in the Cartesian product $X \times \mathbb{R}$ <mark>lying on or above</mark> the function’s graph.

> A **strict** epigraph $\text{epi}_S f$ is the set of points in $X \times \mathbb{R}$ lying _strictly_ above its graph.