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

Comments

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]).

References

[a1]	M. De Wilde, "Closed graph theorems and webbed spaces" , Pitman (1978)
[a2]	H.H. Schaefer, "Topological vector spaces" , Macmillan (1966)