Functions and continuity definition of the number π

The concept of a function is perhaps the most basic one in mathematical analysis. The objects of interest in our subject canoften be represented as functions, and the “ unknowns” in our equations are frequently functions.Therefore, we will spend some time developing and understanding various kinds of functions,including functions defined by polynomials, by power series, and as limits of other functions.In particular, we introduce in this chapter the elementary transcendental functions. We begin with the familiar set theoretical notion of a function,and then move quickly to their analytical properties, specifically that of continuity.

The main theorems of this chapter include:

1. The Intermediate Value Theorem ( [link] ),
2. the theorem that asserts that a continuous real-valued function on a compact set attains a maximum and minimum value ( [link] ),
3. A continuous function on a compact set is uniformly continuous ( [link] ),
4. The Identity Theorem for Power Series Functions ( [link] ),
5. The definition of the real number $\pi ,$
6. The theorem that asserts that the uniform limit of a sequence of continuous functions is continuous ( [link] ), and
7. the Weierstrass $M$ -Test ( [link] ).