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 [...].