Intermediate_Value_Theorem

[!warning] AI-Generated This file was generated by AI and may require review.

Definition

If $f:[a,b] \rightarrow \mathbb{R}$ is continuous, and if $L$ is a real number satisfying $f(a) < L < f(b)$ or $f(b) < L < f(a)$, then there exists a point $c \in (a,b)$ where $f(c) = L$.

In other words, a continuous function on a closed interval attains every value between $f(a)$ and $f(b)$.

Proof

Without loss of generality, assume $f(a) < L < f(b)$. Define the set

$$ K = \{x \in [a,b] : f(x) < L\} $$

Since $a \in K$, the set is nonempty. Since $K \subseteq [a,b]$, it is bounded above. Let $c = \sup K$.

By continuity, if $f(c) < L$, we could find a neighborhood where $f(x) < L$, contradicting that $c$ is the supremum. Similarly, if $f(c) > L$, we could find a neighborhood where $f(x) > L$, contradicting that $c$ is an upper_bound for $K$.

Therefore $f(c) = L$.