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:$$\begin{aligned}\lim_{x \to \infty} \ln\left(\dfrac{x+1}{x}\right)&= \ln\left(\lim_{x \to \infty} \dfrac{x+1}{x}\right) && (\text{since } \ln \text{ is continuous on } (0,\infty)) \\ &= \ln\left(\lim_{x \to \infty} \left(1+\frac{1}{x}\right)\right) && (\text{by algebraic simplification}) \\ &= \ln(1+0) \\ &= \ln(1) \\ &= 0.\end{aligned}$$

Continuity and Differentiability

Theorem Continuity and Differentiability

If a function $f$ is differentiable at a point $a$, then $f$ is continuous at $a$.
If a function $f$ is differentiable on an interval $I$, then $f$ is continuous on $I$.

Note

The converse of this theorem is false. A function can be continuous at a point without being differentiable there.
The classic example is the absolute value function $f(x) = |x|$ at $x=0$:

It is continuous at 0 because $\displaystyle\lim_{x \to 0} |x| = 0 = f(0)$.
It is not differentiable at 0 because the slope on the left is $-1$ and the slope on the right is $+1$.

A continuous function that is not differentiable at $a$ is often characterized by a "corner" or "cusp" in its graph.

Continuity and Sequences

Theorem Fixed Point Theorem

Let $(u_n)$ be a sequence defined by $u_{n+1} = f(u_n)$ that converges to a limit $\ell$.
If the function $f$ is continuous at $\ell$, then the limit $\ell$ is a solution to the equation:$$ f(x) = x $$

Note

The continuity of $f$ at $\ell$ is essential for this theorem to hold.
In practice, we solve $f(x)=x$ to find potential limits and then use sequence properties to determine which one is correct.

Continuity and Equations

Theorem Intermediate Value Theorem (IVT)

Let $f$ be a continuous function on an interval $[a, b]$.
For every real number $k$ between $f(a)$ and $f(b)$, there exists at least one solution $c$ in the interval $[a, b]$ such that:$$ f(c) = k $$

Theorem Corollary: Bijection Theorem

If $f$ is continuous and strictly monotonic (always increasing or always decreasing) on $[a, b]$, then for every $k$ between $f(a)$ and $f(b)$, the equation $f(x) = k$ has a unique solution $c$ in $[a, b]$.

Example

The equation $x^3 = 2$ has a unique solution on $]-\infty, +\infty[$ because the function $f(x) = x^3$ is continuous and strictly increasing on $\mathbb{R}$, and 2 is between $\displaystyle\lim_{x \to -\infty} x^3 = -\infty$ and $\displaystyle\lim_{x \to +\infty} x^3 = +\infty$.