Discontinuities and Function Behavior — Before Limits

What is a Discontinuity?

A discontinuity is a place on a graph where the function is not connected smoothly.
Think of it as a place where you have to lift your pencil while drawing the graph.

We say a function is continuous if you can draw its graph without lifting your pencil.
If you do have to lift it, the function is discontinuous at that point.

Types of Discontinuities

We’ll focus on three major types of discontinuities that show up in graphs:

1. Removable Discontinuity (a "hole")

• The graph looks almost perfect — but there’s a hole at one point.
• It’s like the function is missing one value — but everywhere else behaves smoothly.
• This usually happens when a function is defined almost everywhere, except at one point.

Example:

$f(x) = \frac{x^2 - 1}{x - 1} = \frac{(x - 1)(x + 1)}{x - 1}$

This simplifies to $f(x) = x + 1$ except at $x = 1$ (since division by 0 is undefined).

So we get a straight line with a hole at $x = 1$.

2. Jump Discontinuity

• The graph suddenly jumps from one height to another.
• You might see this in piecewise functions that change rules at certain $x$-values.
• The function has a value on both sides of the jump, but they don’t match.

Example:

$f(x) = \begin{cases} 2 & \text{if } x < 0 \\ 5 & \text{if } x \geq 0 \end{cases}$

When $x$ moves past 0, the function jumps from 2 to 5.
The graph has a gap (discontinuity) at $x = 0$.

3. Infinite Discontinuity (vertical asymptote)

• The graph shoots up or down very fast
• There is a vertical asymptote — a vertical line that the graph tries to avoid but gets close to.

Example:

$f(x) = \frac{1}{x}$

Near $x = 0$, the function becomes very large or very small.
It blows up and never actually reaches a value at $x = 0$.

How to Recognize These from Graphs

Why Discontinuities Matter

• Discontinuities tell us where a function changes behavior.
• They are important in real-world modeling: for example, jumps in pricing, holes in data, or boundaries in physics.
• We need to understand them before we study limits and calculus.

Sketch Examples

1. Removable Discontinuity:

2. Jump Discontinuity:

3. Infinite Discontinuity:

Exercises

1. Determine Continuity

a) $log_3(x)$

b)

$$f(x) = \begin{cases} 2x & \text{if } x < 2 \\ x^3 & \text{if } x \geq 2 \end{cases}$$

c)

$$f(x) = \begin{cases} 2^x & \text{if } x < 4 \\ x^2 & \text{if } x \geq 4 \end{cases}$$

d) $\frac{2}{x+3} +2$