1 - Continuity

Summary

Continuity of mappings between Euclidean spaces is the central topic in this chapter. We begin by discussing those properties of the n-dimensional spaceRⁿ that are determined by the standard inner product In particular, we introduce the notions of distance between the points ofRⁿ and of an open set inRⁿ these, in turn, are used to characterize limits and continuity of mappings between Euclidean spaces. The more profound properties of continuous mappings rest on the completeness ofRⁿ, which is studied next. Compact sets are infinite sets that in a restricted sense behave like finite sets, and their interplay with continuous mappings leads to many fundamental results in analysis, such as the attainment of extrema as well as the uniform continuity of continuous mappings on compact sets. Finally, we consider connected sets, which are related to intermediate value properties of continuous functions.

In applications of analysis in mathematics or in other sciences it is necessary to consider mappings depending on more than one variable. For instance, in order to describe the distribution of temperature and humidity in physical space-time we need to specify (in first approximation) the values of both the temperature T and the humidity h at every (x, t) ∈ R³ x R ≃ R⁴, where x ∈ R³ stands for a position in space and t ∈ R for a moment in time. Thus arises, in a natural fashion, a mapping f : R⁴ → R² with f(x, t) = (T, h). The first step in a closer investigation of the properties of such mappings requires a study of the space Rⁿ itself.