Teacher Guide: Types of Discontinuity

Learn to tell removable discontinuities (holes), jump discontinuities and infinite discontinuities apart, from graphs and from formulas.

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Learning Objectives

Define removable, jump and infinite discontinuities using one-sided and two-sided limits
Classify a discontinuity from a graph (hole, gap between pieces, vertical asymptote)
Classify the discontinuities of rational functions by factoring and cancelling
Find the coordinates of a hole and redefine $f(a)$ to remove it
Compute the jump size of a piecewise function

Prerequisites

• Continuity: the three conditions for a function to be continuous at a point
• One-Sided Limits: left-hand and right-hand limits
• Infinite Limits: limits equal to $\infty$ or $-\infty$ and vertical asymptotes
• Limits by Direct Substitution and factoring to resolve $\frac{0}{0}$
• Factoring quadratic expressions

Discussion Starters

1. Why does the value of $f(a)$ not change the type of a discontinuity?
2. A friend says every zero of the denominator gives a vertical asymptote. How would you convince them otherwise?
3. Where in everyday life do prices or quantities jump suddenly instead of changing smoothly?
4. Could a function have a jump discontinuity and a hole at the same $x$ -value? Why or why not?

Common Misconceptions

Every zero of the denominator is a vertical asymptote

Remediation: Graph $\frac{x^2 - x - 2}{x - 2}$ and $\frac{1}{x - 2}$ side by side. Ask students to factor both and find what is different. Then state the rule: cancel first, then decide.

A filled dot at $x = a$ means the function is continuous there

Remediation: Show the jump example $h(1) = 6$ : the dot is filled, yet the left side approaches $3$ . Continuity needs the limit to exist and equal $f(a)$ , not just a defined value.

The jump size is the difference between $f(a)$ and one side

Remediation: Define the jump size only with the two one-sided limits and have students compute it for a function where $f(a)$ is undefined.

Differentiation Ideas

For Struggling Students:

• Start with graphs only: circle holes, mark gaps between pieces, draw dashed asymptotes
• Use tables of values near $x = a$ to see whether outputs settle, split, or blow up
• Give a decision chart: limit exists? sides equal? values blow up?

For On-Level Students:

• Classify discontinuities of rational functions by factoring
• Find hole coordinates and jump sizes
• Write a piecewise function with a prescribed jump

For Advanced Students:

• Find a parameter $k$ that turns a jump into a continuous join
• Explore oscillating discontinuities like $\sin\left(\frac{1}{x}\right)$
• Build a rational function with a hole at $(1, 2)$ and an asymptote at $x = -3$

Standards Alignment

HSF-IF.C.7d (CCSS.MATH.CONTENT.HSF.IF.C.7.D)
Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior
HSF-IF.C.7b (CCSS.MATH.CONTENT.HSF.IF.C.7.B)
Graph square root, cube root, and piecewise-defined functions, including step functions
HSA-APR.D.6 (CCSS.MATH.CONTENT.HSA.APR.D.6)
Rewrite simple rational expressions in different forms

Lesson Resources

visualThree Graphs, Three Types
Interactive graphs of a hole, a jump and an asymptote with tables of values near the break
activityDiscontinuity Sorting Cards
Students sort 12 graphs and formulas into removable, jump and infinite
worksheetHoles or Asymptotes?
Factor rational functions, locate holes with coordinates and vertical asymptotes

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

A function

f

is continuous at

x = a

when

\lim_{x \to a} f(x) = f(a)

. If this fails,

f

has a discontinuity at

x = a

. How it fails tells you the type:

1. Removable discontinuity (hole) The limit exists and is a finite number

L

, but

f(a)

is missing or

f(a) \neq L

. The graph has a single hole at

(a, L)

. You can remove it by redefining

f(a) = L

2. Jump discontinuity Both one-sided limits are finite but different:

\lim_{x \to a^-} f(x) \neq \lim_{x \to a^+} f(x)

The graph jumps from one height to another. The jump size is

\left| \lim_{x \to a^+} f(x) - \lim_{x \to a^-} f(x) \right|

. No single value of

f(a)

can fix it.

3. Infinite discontinuity At least one one-sided limit is

\infty

-\infty

. The graph has a vertical asymptote

x = a

From a formula (rational functions): factor the numerator and the denominator.

A factor $(x - a)$ that cancels completely gives a hole at $x = a$ .
A factor $(x - a)$ that stays in the denominator gives a vertical asymptote at $x = a$ .

Worked Examples

Classify the discontinuity of $f(x) = \frac{x^2 - x - 2}{x - 2}$ at $x = 2$ and remove it.

Check where $f$ is undefined

The denominator $x - 2$ is $0$ at $x = 2$ → $f(2)$ is undefined

Factor the numerator

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

Cancel the common factor

$f(x) = x + 1$ for every $x \neq 2$ → The factor $(x - 2)$ cancels completely

Find the limit

$\lim_{x \to 2} f(x) = 2 + 1$ → $= 3$ , a finite limit

Classify and fix

Limit exists, $f(2)$ does not: a hole at $(2, 3)$ . Define $f(2) = 3$ . → Removable discontinuity

Answer: Removable discontinuity (hole) at $(2, 3)$ . Setting $f(2) = 3$ makes $f$ continuous.

Common Mistakes

Drawing a vertical asymptote at every zero of the denominator

Why it's wrong: A zero of the denominator can also be a hole. In $\frac{x^2 - x - 2}{x - 2}$ the factor $(x - 2)$ cancels, so there is no asymptote at $x = 2$ .

Correct: Factor and cancel first. Only factors left in the denominator give vertical asymptotes; cancelled factors give holes.

Calling a jump removable because both one-sided limits exist

Why it's wrong: Removable needs the two-sided limit, which exists only when both sides agree. At a jump they are different numbers.

Correct: Equal one-sided limits: removable (or continuous). Different finite one-sided limits: jump.

Using $f(a)$ as the height of the hole

Why it's wrong: At a hole $f(a)$ is missing or wrong. The hole sits at the height of the limit.

Correct: Find the $y$ -coordinate of a hole by substituting $x = a$ into the simplified function: the hole of $\frac{x^2 - 9}{x^2 - x - 6}$ is at $\left(3, \frac{6}{5}\right)$ .

Thinking a cancelled factor always means a hole, even when it appears twice in the denominator

Why it's wrong: In $\frac{x - 1}{(x - 1)^2}$ only one factor cancels; $\frac{1}{x - 1}$ is left, which still blows up at $x = 1$ .

Correct: Cancel as far as possible, then check: if $(x - a)$ is still in the denominator, it is an infinite discontinuity.

Why It Matters

Knowing the type of a discontinuity tells you how serious the break is and what you can do about it:

Holes are harmless gaps: the function is predictable around them, and one redefined point makes it continuous. Computer programs that crash on $\frac{0}{0}$ are often fixed exactly this way.
Jumps describe sudden changes: parking fees, shipping rates, tax brackets and switches that turn on or off.
Infinite discontinuities warn that a quantity blows up: lens images moving to infinity, resonance in bridges, or a formula dividing by something close to zero.

In calculus, the type of discontinuity decides whether a limit exists, whether a function can be integrated over an interval, and how a graph must be drawn. Graphing rational functions correctly (holes versus asymptotes) is one of the most tested skills before calculus.

Real World Applications

Parking Fees (Jumps)

Parking garages charge by the started hour, so the price function is constant on each hour and jumps at every full hour.

Example:

A garage charges 4 dollars for up to 1 hour and 7 dollars for more than 1 hour up to 2 hours. At $t = 1$ the price jumps by 3 dollars.

1Try It Yourself

The same garage charges 7 dollars for more than 1 hour up to 2 hours and 10 dollars for more than 2 hours up to 3 hours. $C(t)$ is the price for $t$ hours.

What is the size of the jump in the price at $t = 2$ hours (in dollars)?

Step 1: Write the mathematical expression

Subtract the left-hand limit from the right-hand limit:

Lenses and Images at Infinity

For a lens with focal length $10$ cm, an object at distance $u$ produces an image at distance $v(u) = \frac{10u}{u - 10}$. When the object reaches the focal point, the image moves off to infinity.

Example:

At $u = 15$ cm the image is at $v = \frac{150}{5} = 30$ cm. At $u = 10.1$ cm it is already at $v = \frac{101}{0.1} = 1010$ cm.

2Try It Yourself

The image distance is $v(u) = \frac{10u}{u - 10}$ (in cm).

At which object distance $u$ (in cm) does $v$ have an infinite discontinuity?

Step 1: Write the mathematical expression

Find where the denominator is zero and the numerator is not:

Fixing a Division by Zero in Software

Programs often compute an average rate of change $\frac{f(t) - f(a)}{t - a}$. At $t = a$ the formula gives $\frac{0}{0}$ and the program crashes, although the values nearby are perfectly smooth: a removable discontinuity.

Example:

For $f(t) = t^2$ and $a = 3$ , the rate $\frac{t^2 - 9}{t - 3}$ simplifies to $t + 3$ , so the programmer stores the value $6$ for $t = 3$ .

3Try It Yourself

A program computes $R(t) = \frac{t^2 - 16}{t - 4}$ and crashes at $t = 4$ .

Which value should the programmer assign to $R(4)$ so that $R$ is continuous?

Step 1: Write the mathematical expression

Simplify, then substitute $t = 4$ :

Key Takeaways

1A discontinuity at $x = a$ means $\lim_{x \to a} f(x) = f(a)$ fails
2Removable (hole): the limit $L$ exists, but $f(a)$ is missing or different; redefine $f(a) = L$ to fix it
3Jump: finite one-sided limits that differ; jump size $= \left| \lim_{x \to a^+} f(x) - \lim_{x \to a^-} f(x) \right|$
4Infinite: a one-sided limit is $\pm\infty$ ; the graph has a vertical asymptote $x = a$
5For rational functions, factor first: cancelled factors give holes, factors left in the denominator give asymptotes
6The $y$ -coordinate of a hole comes from the simplified function, never from $f(a)$

Frequently Asked Questions

Why is a hole called removable?

Because one point is enough to repair it. The limit

L

already exists, so defining (or redefining)

f(a) = L

makes

\lim_{x \to a} f(x) = f(a)

true, and the function becomes continuous at

a

Can a jump discontinuity ever be removed?

No. The left side heads to one height and the right side to another. Whatever value you give

f(a)

, it can match at most one side, so the two-sided limit still does not exist.

Does a function value at $x = a$ change the type?

No. The type depends only on the limits near

a

. A jump stays a jump whether

f(a)

is the left value, the right value or undefined, and a hole stays removable whether

f(a)

is missing or wrong.

Are there other types of discontinuity?

Yes, but they are rarer. For example

\sin\left(\frac{1}{x}\right)

oscillates faster and faster near

x = 0

, so it has no one-sided limits at all (an oscillating discontinuity). Jumps, infinite and oscillating discontinuities are all non-removable; many books call only the infinite and oscillating ones essential.

Glossary

Discontinuity: A point $x = a$ where $\lim_{x \to a} f(x) = f(a)$ does not hold
Removable discontinuity (hole): The limit exists and is finite, but $f(a)$ is undefined or not equal to it; fixed by redefining one point
Jump discontinuity: Both one-sided limits are finite but not equal
Jump size: The distance between the two one-sided limits: $\left| \lim_{x \to a^+} f(x) - \lim_{x \to a^-} f(x) \right|$
Infinite discontinuity: At least one one-sided limit is $\infty$ or $-\infty$ ; the graph has a vertical asymptote
Vertical asymptote: A vertical line $x = a$ that the graph approaches while its values grow without bound

Formula Card

Removable (hole)

\lim_{x \to a} f(x) = L

exists,

f(a) \neq L

or undefined

Fix it with $f(a) = L$

Jump

\lim_{x \to a^-} f(x) \neq \lim_{x \to a^+} f(x)

, both finite

Jump size $= \left| \lim_{x \to a^+} f(x) - \lim_{x \to a^-} f(x) \right|$

Infinite

\lim_{x \to a^\pm} f(x) = \pm\infty

Vertical asymptote $x = a$

Rational function test

\frac{(x - a)\,p(x)}{(x - a)\,q(x)} = \frac{p(x)}{q(x)}

A cancelled factor gives a hole at $\left(a, \frac{p(a)}{q(a)}\right)$ if $q(a) \neq 0$; a factor left in the denominator gives an asymptote

Teacher Guide: Types of Discontinuity

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Worked Examples

Common Mistakes

Why It Matters

Real World Applications

Parking Fees (Jumps)

Lenses and Images at Infinity

Fixing a Division by Zero in Software

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Removable (hole)

Jump

Infinite

Rational function test

More in This Topic

Introduction to Limits

Limit Notation

Evaluating Limits Graphically

Evaluating Limits from Tables