C-10 - Applications of the Baire Category Theorem to Real Analysis

Abstract

This chapter discusses the applications of the Baire Category theorem (BCT) related to real-valued functions. The spaces under consideration are metric spaces. The BCT has numerous applications in the setting of real-valued functions of a real variable. Some uses involve showing that a condition satisfied pointwise, is actually satisfied globally. Another example of this type of use of the BCT implies that if a function ƒ : ℝ→ ℝ has the property that for each x ∈ ℝ there exists n given by n(x) ∈ ℕ such that ƒ(n)(x) is 0, then there exists N ∈ ℕ such that ƒ(N)(x) is 0 for every x ∈ ℝ, and so ƒ is a polynomial. An important type of application of the BCT involves its use in establishing the existence of certain objects that might be difficult to visualize or construct. A classical example is the use of the BCT to show the existence of continuous nowhere differentiable functions (that is, functions that at no point have a finite derivative).This chapter focuses on such structure theorems for several familiar metric spaces that arise in the theory of real-valued functions.