Lesson 43 — Continuity of Functions

Definitions and theorems

Direct ε-δ form

$f$ continuous at $a$ $\iff$ $\forall \eps > 0, \exists \delta > 0 : |x - a| < \delta \Rightarrow |f(x) - f(a)| < \eps$.

(Note: here we may have $|x - a| = 0$, that is, $x = a$, since $|f(a) - f(a)| = 0 < \eps$ trivially.)

Types of discontinuity

Intermediate Value Theorem (IVT)

If $f \in C([a, b])$ and $k$ is between $f(a)$ and $f(b)$, then $\exists c \in (a, b) : f(c) = k$.

Weierstrass theorem (extrema)

If $f \in C([a, b])$ (closed and bounded interval), then $f$ attains a maximum and minimum on $[a, b]$.

Uniform continuity

$f$ is uniformly continuous on $I$ if $\forall \eps, \exists \delta$ (the same $\delta$ for every $x$) such that $|x - y| < \delta \Rightarrow |f(x) - f(y)| < \eps$.

Heine-Cantor: $f \in C([a, b])$ ⇒ $f$ uniformly continuous.

Operations preserving continuity

• Sum, product, quotient (denom. $\neq 0$).
• Composition: $f \circ g$ continuous if $g$ continuous at $a$ and $f$ at $g(a)$.
• $\max(f, g)$, $\min(f, g)$, $|f|$.