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Math and science::Analysis::Tao::09. Continuous functions on R

The intermediate value theorem

Continuous functions whose domain is closed enjoy two useful properties:

  • the maximum principle
  • the intermediate value theorem

This card covers the second.

Intermediate value theorem (my version)

Let \( a < b \) be reals, let \( X = [a, b] \), and let \( f: X \to \mathbb{R} \) be a continuous function.

Then for every \( y \in [f_{min}, f_{max}] \), where \( f_{min} \) and \( f_{max} \) are the minimum and maximum obtained by \( f \) (which exist by the maximum principle), there is [...] such that [...].