Continuous Functions

Abstract

Let $I \subseteq \mathbf{R}$ be an interval—open, closed, or otherwise. For the sake of simplicity, but without great loss, we mainly consider functions f: I → R. Roughly speaking, if f is continuous then f(x) is close to f(x ₀) whenever x ∈ I is close to x ₀ ∈ I. Many functions which arise naturally in applications are continuous on some interval I. We shall see that continuous functions have very nice properties. The two big theorems in the world of continuous functions are the Intermediate Value Theorem and the Extreme Value Theorem . We prove these using bisection algorithms.

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