Continuity

at a point

A function $f : 𝒟 \to ℛ$ is continuous at $ω$ if

$f$ is defined at $ω$ i.e., $ω \in 𝒟$ ;
$\lim_{x \to ω} f (x)$ exists;
$\lim_{x \to ω} f (x) = f (ω)$ .

One sided

Right continuous: $\lim_{x \to ω^{+}} f (x) = f (ω)$
Left continuous: $\lim_{x \to ω^{-}} f (x) = f (ω)$

on interval

A function is continuous on an interval $ℐ$ if it is continuous at every point in $ℐ$

Continuity properties

$𝒶 f (x) \pm 𝒷 g (x)$
$f (x) g (x)$
$\frac{f (x)}{g (x)}$ when defined
${f (x)}^{n}$ for $n \in ℤ^{+}$
$\sqrt[n]{f (x)}$ , $n \in ℤ^{+}$ when defined

Polynomial and rational functions are continuous on their domain.

Trigonometric functions are continuous on their domains.

Inverse functions of continuous functions are continuous on their domain.

Discontinuities

removable: $\lim_{x \to ω} f (x) \in ℝ$
jump: $\lim_{x \to ω^{+}} f (x) \in ℝ, \lim_{x \to ω^{-}} f (x) \in ℝ$ and $\lim_{x \to ω^{+}} f (x) \neq \lim_{x \to ω^{-}} f (x)$

Functions with removable discontinuities have continuous extensions.

Discontinuities

infinite: $\lim_{x \to ω^{+}} f (x) = \pm \infty$ or $\lim_{x \to ω^{-}} f (x) = \pm \infty$
oscillating: $\lim_{x \to ω} f (x)$ approaches more than one value

Explanations and Examples

Find the interval of continuity

$f_{1} (x) = \frac{x - 1}{x^{2} + 2 x}$
$f_{2} (x) = \sqrt{x^{2} + 2 x - 5}$
$f_{3} (x) = sin (x^{2} - 4)$

Evaluate the limits

$lim_{x \to \frac{π}{2}} cos (2 x + sin (\frac{3 π}{2} + x))$
$lim_{x \to 0} \sqrt{x + 1} ℯ^{tan x}$

Show that $p (x) = x^{3} + 3 x^{2} + x - 1$ has a root between zero and one.

Let $f (x) = \frac{| x |}{x}$ , then $f (-3) = 1$ and $f (2) = 1$ , does that the Intermediate Value Theorem imply there is a root between negative three and two?

Implications

Composition of continuous functions

If $f$ is continuous at $ω$ and $g$ is continuous at $f (ω)$ then $f \circ g$ is continuous at $ω$

If $\lim_{x \to ω} f (x) = 𝓁$ and $g (x)$ is continuous at $𝓁$ then

\lim_{x \to ω} g (f (x)) = g (\lim_{x \to ω} f (x))

Intermediate value theorem

Let $f (x)$ be continuous on an interval $[a, b]$ then.

\forall ν \in (𝓂, ℳ), \exists ω \in (a, b) : f (ω) = ν

where

$𝓂 = \min (f (a), f (b)),$ and
$ℳ = \max (f (a), f (b))$