# Graph of a function

##### Article

July 3, 2022 The graph of a function f from E into F is the subset G of E×F formed by the pairs of elements linked by the correspondence: G { ( x , there ) ∈ E × F ∣ there f ( x ) } { ( x , f ( x ) ) ∣ x ∈ E } . {\displaystyle G\{(x,y)\in E\times F\mid yf(x)\}\{(x,f(x))\mid x\in E\}.}

## Numerical functions

This set is called the graph of f because it makes it possible to give a graphical representation of it in the usual case where E and F are sets of real numbers: indeed, we can then sometimes represent E and F on two intersecting axes, each pair of G can then be represented by a point in the plane, provided with a reference defined by the two axes. We also speak of a curve representative of the function. If E is the plane ℝ2 and F is the set of real numbers ℝ, the graph of the function is a gauche surface in 3-dimensional Euclidean space. It is then possible to reduce to a plane representation by considering level curves, that is to say by drawing in the starting plane an altimetric map of the relief of the left surface. In the case of complex functions, E is the complex plane C and F is also the set of complexes C. The need for 4 dimensions makes the graphical representation more complicated. Several methods exist, either by using two 3-dimensional graphs (real and imaginary parts, modulus and argument), or by using a 2-dimensional graph associated with the coloring of regions.

## Tests of verticals and horizontals

Vertical line test A subset G of E×F is the graph of a function from E to F if and only if for any element x of E, G∩({x}×F) is a singleton or empty. It is the graph of a map from E to F if and only if for all x in E, G∩({x}×F) is a singleton. Horizontal line test A function from E to F with graph G is injective if and only if for any element y of F, G∩(E×{y}) is a singleton or empty. It is surjective if and only if for all y in F, G∩(E×{y}) is nonempty. A subset G of E×F is therefore the graph of a bijection from E to F if and only if for any x in E, G∩({x}×F) is a singleton and for any y in F, G∩(E×{y}) is a singleton.

## Topology

When E and F are topological spaces, F being separate, if the map f is continuous then its graph is closed in E×F. The converse is false, as evidenced by the application of ℝ in ℝ which associates 0 with x if x ≤ 0 and 1/x if x > 0. It is however true if F is compact (or even only quasi-compact). These two implications generalize to multivalued functions.