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  "body": "\n## Summary\n\nThe epigraph of a function is the set of points lying on or above its graph.\nIt turns statements about functions into statements about sets.\nA function is [[convex|07Q7CNM5WF]] exactly when its epigraph is a convex set, and lower [[semicontinuous|RN4G8SGP6P]] exactly when its epigraph is closed.\nThis dictionary is the starting point of convex analysis and variational analysis.\nThe [[definition|5Q9BNSZ5XZ]] gives the precise set.\n\n## Explanations\n\n### L1 human\n\nDraw the graph of a function and shade everything above it: the shaded region is the epigraph.\nFor $f(x) = x^2$ it is the inside of the upward parabola, together with the parabola itself.\nA bowl-shaped function has a shaded region with no dents, which is what convexity of the set means.\nA jump in the graph shows up as a missing or present edge of the region, which is how semicontinuity is seen.\n",
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