| # Graph, epigraph, level and sublevel sets of a function Consider a function . We can define four sets relevant to : the graph and the epigraph, both are subsets of . Level and sub-level sets are subsets of . ## Graph The graph of is the set of input-output pairs that can attain, that is: ## Epigraph The epigraph, denoted , describes the set of input-output pairs that can achieve, as well as ‘‘anything above’’ (epi in Greek means ‘‘above’’):
| The function , with domain and value inside the domain . The graph corresponds to the points in blue, and the epigraph is in light blue. The epigraph extends to infinity above the graph. Points outside the domain are not shown.
## Level sets Level sets are sets of points that achieve exactly a certain value for . Precisely, the -level set of is defined by In two-dimensions (), the level sets are referred to as level curves. ## Sub-level sets A related notion is that of sub-level set. This is now the set of points that achieve at most a certain value for , or below. Precisely, the -sub-level set of is defined by
| Level and sub-level sets of a function .