---
tag: 8BJ5M76FSX
kind: concept
name: Epigraph
msc: [52A41, 26B25, 49J53]
sources:
  - url: https://en.wikipedia.org/wiki/Epigraph_(mathematics)
same_as:
  wikidata: Q1347059
  wikipedia: Epigraph (mathematics)
---

## Summary

The epigraph of a function is the set of points lying on or above its graph.
It turns statements about functions into statements about sets.
A function is [[convex|07Q7CNM5WF]] exactly when its epigraph is a convex set, and lower [[semicontinuous|RN4G8SGP6P]] exactly when its epigraph is closed.
This dictionary is the starting point of convex analysis and variational analysis.
The [[definition|5Q9BNSZ5XZ]] gives the precise set.

## Explanations

### L1 human

Draw the graph of a function and shade everything above it: the shaded region is the epigraph.
For $f(x) = x^2$ it is the inside of the upward parabola, together with the parabola itself.
A bowl-shaped function has a shaded region with no dents, which is what convexity of the set means.
A jump in the graph shows up as a missing or present edge of the region, which is how semicontinuity is seen.