6. Continuity

This chapter takes us back to general real valued functions.

We will focus here on those functions whose value at a single point is influenced by the function's behaviour in the neighbourhood of that point, providing the function with a 'continuous' character. This kind of interaction is no feature of the original notion of a function, so we expect new properties for these functions.

As we need to examine the function's behaviour in an arbitrary neighbourhood of a fixed piont, we will employ convergent sequences for this study.

The evolution of continuity has been a long term process. Rudiments are seen from L. Euler, but concepts that match our modern ideas are first due to A. L. Cauchy and B. Bolzano. B. Riemann provided important examples of continuous functions with special properties.

A. L. Cauchy

B. Riemann

  1. Image Sequences
  2. Continuous Functions
  3. Calculation Rules for Continuous Functions
  4. Properties of Continuous Functions
  5. Uniformly Continuous Functions
  6. Continuous Functions on Closed Intervals
  7. The Weierstrass Approximation Theorem
  8. Continuously Extendable Functions


  9. Properties of Continuously Extendable Functions
  10. Sequences of Continuous Functions