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 an \( x \in [a, b] \) such that \( f(x) = y \) .
Intermediate value theorem (standard version)
Let \( a < b \) be reals, let \( X = [a, b] \), and let \( f: X \to \mathbb{R} \) be a continuous function. Let \( y \) be a real between \( f(a) \) and \( f(b) \), in other words, \( min(f(a), f(b)) \ge y \ge max(f(a), f(b)) \).
Then there exists a \( c \in [a, b] \) such that \( f(c) = y \).
Proof: TODO