Graph, epigraph, level and sublevel sets of a function

Consider a function $f : mathbf{R}^n rightarrow mathbf{R}$ . We can define four sets relevant to : the graph and the epigraph, both are subsets of $mathbf{R}^{n+1}$ . Level and sub-level sets are subsets of $mathbf{R}^n$ .

Graph

The graph of is the set of input-output pairs that can attain, that is:

$G(f) := left{ (x,f(x)) in mathbf{R}^{n+1} ~:~ x in mathbf{R}^n right}.$

Epigraph

The epigraph, denoted mbox{bf epi} f , describes the set of input-output pairs that can achieve, as well as ‘‘anything above’’ (epi in Greek means ‘‘above’’):

$mbox{bf epi} f := left{ (x,t) in mathbf{R}^{n+1} ~:~ x in mathbf{R}^n, ;; t ge f(x) right}.$

The function f : mathbf{R} rightarrow mathbf{R} , with domain (-1,1) and value inside the domain f(x) = x^2 + (1/2) sin (10x) . 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

$L_t(f) := left{ xin mathbf{R}^{n} ~:~ x in mathbf{R}^n, ;; t = f(x) right} .$

In two-dimensions ( n=2 ), 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

$S_t(f) := left{ xin mathbf{R}^{n} ~:~ x in mathbf{R}^n, ;; t ge f(x) right} .$

Level and sub-level sets of a function $f : mathbf{R}^2 rightarrow mathbf{R}$ .