epigraph

The epigraph of f:\mathbb R^n\to\mathbb R is \operatorname{epi}(f)=\{(x,t)\in\mathbb R^n\times\mathbb R : t\ge f(x)\}. A function is convex exactly when its epigraph is a convex set, which lets every set-convexity tool apply directly to functions: a bowl-shaped graph's epigraph is a genuinely convex filled region, while a wavy graph's epigraph has a dent exactly where the wave crests and dips back down.