Continuity

Definition

Definition Continuity at a Point

A function $f$ is continuous at a point $x=a$ if three conditions are met:

$f(a)$ is defined (the point exists).
$\displaystyle\lim_{x \to a} f(x)$ exists (the limit exists).
$\displaystyle\lim_{x \to a} f(x) = f(a)$ (the limit equals the function's value).

Intuitively, a function is continuous over an interval if its graph can be drawn without lifting your pen from the paper.

Types of Discontinuity

Removable Discontinuity: Occurs when the limit exists but is not equal to the function's value, or the function is not defined at the point. This corresponds to a "hole" in the graph. It can be "removed" by redefining the function at that single point.
Non-Removable (Essential) Discontinuity: Occurs when the limit does not exist. This corresponds to a "jump" (where one-sided limits differ), a vertical asymptote (an infinite discontinuity), or more complicated oscillatory behaviour.

Proposition A Catalog of Continuous Functions

The following types of functions are continuous at every number in their domains:

Polynomials (e.g., $f(x)=x^2-3x+5$)
Rational functions (e.g., $f(x)=\dfrac{x+1}{x-2}$, continuous for $x \neq 2$)
Root functions (e.g., $f(x)=\sqrt[n]{x}$, continuous on their domains)
Trigonometric functions (e.g., $\sin(x), \cos(x), \tan(x)$, etc., continuous on their domains)
Inverse trigonometric functions (e.g., $\arctan(x), \arcsin(x)$, etc.)
Exponential functions (e.g., $f(x)=e^x$)
Natural logarithm (e.g., $f(x)=\ln(x)$, continuous for $x>0$)

Furthermore, any sum, difference, product, or composition of these functions is also continuous on its domain.

Limit of a Composite Function

Proposition Limit of a Composite Function

If $\displaystyle\lim_{x \to a} g(x) = L$ and if the function $f$ is continuous at $L$, then:$$ \lim_{x \to a} f(g(x)) = f\left(\lim_{x \to a} g(x)\right) = f(L). $$

In essence, if the outer function is continuous, you can "move the limit inside the function."

Example

Evaluate $\displaystyle\lim_{x \to \infty} \ln\left(\dfrac{x+1}{x}\right)$.

Answer

We apply the Limit of a Composite Function rule. Since the natural logarithm function is continuous for all positive inputs, we can move the limit inside the function:$$\begin{aligned}\lim_{x \to \infty} \ln\left(\dfrac{x+1}{x}\right)&= \ln\left(\lim_{x \to \infty} \dfrac{x+1}{x}\right) && (\text{since } \ln \text{ is continuous on } (0,\infty)) \\ &= \ln\left(\lim_{x \to \infty} \left(1+\frac{1}{x}\right)\right) && (\text{by algebraic simplification}) \\ &= \ln(1+0) \\ &= \ln(1) \\ &= 0.\end{aligned}$$