On Continuity

A seminar talk, in the academic layout

§ 1

Definitions

Continuity

Definition 1.1 (Continuity)

A function $f: X \to Y$ is continuous at $x_0$ if for every $\varepsilon > 0$ there exists $\delta > 0$ such that $|x - x_0| < \delta$ implies $|f(x) - f(x_0)| < \varepsilon$ .

The Extreme Value Theorem

Theorem 1.2 (EVT)

Every continuous function on a compact set attains its maximum.

Proof

By compactness, $f(K)$ is compact in $\mathbb{R}$ , hence closed and bounded — so $\sup f(K) \in f(K)$ . $\blacksquare$

On Continuity

A seminar talk, in the academic layout

Continuity

The Extreme Value Theorem

Q&A

academic + dark