# Graph of a mapping

$f: X \rightarrow Y$ of a set $X$ into a set $Y$

The subset $\Gamma$ of the product $X \times Y$ consisting of the points $( x, f ( x))$, $x \in X$. If $X$ and $Y$ are topological spaces, $f$ is a continuous mapping and $p: X \times Y \rightarrow X$ is the projection of the topological product $X \times Y$ onto the factor $X$, then the mapping $p$ is a homeomorphism of the subspace $\Gamma$ onto $X$. If $Y$ is a Hausdorff space, then the set $\Gamma$ is closed in the product $X \times Y$.

B.A. Pasynkov

In the case of a real-valued function $f$ of $n$ real arguments $x _ {1} \dots x _ {n}$ and domain of definition $E ^ {n}$, its graph is the set of all ordered pairs $(( x _ {1} \dots x _ {n} ), f ( x _ {1} \dots x _ {n} ))$, where $( x _ {1} \dots x _ {n} )$ is any point of $E ^ {n}$; in other words, it is the set of all points $( x _ {1} \dots x _ {n} , f ( x _ {1} \dots x _ {n} ))$ in $E ^ {n} \times \mathbf R$. Having chosen a coordinate system (Cartesian, polar or any other coordinates), the numerical points $( x, f ( x))$, $( x, y, f ( x, y))$ can be represented by points of the plane or space. For real-valued functions $f$ in one real variable which have derivatives $f ^ { \prime }$, $f ^ { \prime\prime }$, in more or less complicated examples the graph can be sketched by studying the signs of $f ^ { \prime }$ and $f ^ { \prime\prime }$. The sign of $f ^ { \prime }$ is an indicator of the monotony of $f$, while the sign of $f ^ { \prime\prime }$ indicates the direction of convexity of the graph of the function. To obtain an idea on the graph of a real-valued function $z$ in two real variables, the method of sections may be employed: One studies the sections of the graph by certain planes, in particular by planes $z = c$; the projection of this section on the $xy$- plane is said to be a level set of $z$. Similarly, for a function $f$ defined in a domain $E ^ {n}$, the level set of $f$ at level $c$, where $c$ is an arbitrary number, is the set of all solutions of the equation $c = f ( x _ {1} \dots x _ {n} )$. The solutions $( x _ {1} \dots x _ {n} )$ must be found in $E ^ {n}$. A level set may prove to be empty. If a level set is a line or a surface, it is known as a level line or a level surface of the function.

A.A. Konyushkov

An extremely important theorem in functional analysis is the so-called closed-graph theorem: If $X$ and $Y$ are Fréchet spaces (cf. Fréchet space) and $f : X \rightarrow Y$ is a linear mapping with a closed graph, then $f$ is continuous. Many generalizations of this result are known (see [a1]).