Discrete vs Continuous Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Discrete vs Continuous.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The distinction between quantities that take separate, distinct values (discrete, like number of students) and quantities that can take any value in a range (continuous, like height or temperature).

People come in whole numbers (discrete). Height can be any value (continuous).

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Discrete data lands on separate values; continuous data can take any value in a range.

Common stuck point: The procedure for discrete vs continuous is the easy part; the trap is labeling by how a value is reported. Asking "Could the quantity take a value strictly between two adjacent readings? Yes is continuous, no is discrete." first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Could the quantity take a value strictly between two adjacent readings? Yes is continuous, no is discrete.

Worked Examples

Example 1

easy
Classify each quantity as discrete or continuous, and explain: (a) number of students in a class, (b) the height of a student, (c) the number of text messages sent, (d) the temperature in a room.

Answer

Discrete: (a) and (c); Continuous: (b) and (d).

First step

1
(a) Number of students: must be a whole number (can't have 0.50.5 students). Discrete.

Full solution

  1. 2
    (b) Height: can take any value in a range (e.g., 162.7162.7 cm, 162.73162.73 cm, etc.). Continuous.
  2. 3
    (c) Number of texts: whole numbers only. Discrete.
  3. 4
    (d) Temperature: can take any real value in a range (e.g., 20.1°20.1°, 20.15°20.15°, etc.). Continuous.
Discrete quantities are counted (whole numbers, gaps between values), while continuous quantities are measured (any real value possible in an interval). The distinction matters for choosing the right mathematical model and type of graph.

Example 2

medium
A discrete model counts bacteria in a culture as B(t)=2tB(t) = 2^t (where tt is in hours, integer values). A continuous model uses B(t)=e0.693tB(t) = e^{0.693t}. Compare the models at t=0,1,2,3t = 0, 1, 2, 3 and explain when each is appropriate.

Example 3

medium
A study records (a) number of customers per hour and (b) wait time per customer. Classify each and say which chart type fits.

Example 4

medium
A dataset of children's shoe sizes is plotted with a histogram. Why is a bar chart with gaps actually more honest than a histogram here?

Example 5

hard
Argue why money (in dollars and cents) is best modeled as discrete for accounting but often as continuous in financial models.

Example 6

challenge
When modeling a population over time, ecologists sometimes use a discrete recurrence Nt+1=rNt(1−Nt/K)N_{t+1} = rN_t(1 - N_t/K) and other times use a continuous ODE dNdt=rN(1−N/K)\frac{dN}{dt} = rN(1 - N/K). Give one reason each model is preferred in its context.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Which type of graph is appropriate for each: (a) the number of cars sold each month (bar chart or line graph with any real yy-values?), (b) the speed of a car over a journey?

Example 2

medium
Shoe sizes in the UK come in steps of 12\frac{1}{2} (e.g., 6,6.5,7,7.5,…6, 6.5, 7, 7.5, \ldots). Is shoe size discrete or continuous? What about actual foot length in centimetres?

Example 3

easy
Is the number of students in a class discrete or continuous?

Example 4

easy
Is a person's height discrete or continuous?

Example 5

easy
Discrete or continuous: shoe sizes like 8,8.5,98, 8.5, 9?

Example 6

easy
Discrete or continuous: temperature of a room?

Example 7

easy
Discrete or continuous: number of cars in a parking lot?

Example 8

easy
Discrete or continuous: the weight of water in a glass?

Example 9

easy
Discrete or continuous: number of pages in a book?

Example 10

easy
Discrete or continuous: the time elapsed during a race?

Example 11

medium
A factory tracks (a) number of items produced and (b) total weight produced. Classify each.

Example 12

medium
Why is time considered continuous even though clocks display discrete seconds?

Example 13

medium
Classify the variable 'number of heads in 1010 coin flips' and give its possible values.

Example 14

medium
Is money (e.g. dollars and cents) best modeled as discrete or continuous? Explain the nuance.

Example 15

medium
A quantity takes values {0,1,2,3,…}\{0,1,2,3,\ldots\} with no upper bound. Discrete or continuous, finite or infinite?

Example 16

medium
Sketch the difference: graph of points (1,2),(2,4),(3,6)(1,2),(2,4),(3,6) vs the line y=2xy=2x. Which is discrete and which continuous?

Example 17

medium
Classify: shoe sizes are discrete, but foot length is continuous. Reconcile these.

Example 18

medium
A survey records (a) number of pets owned and (b) distance commuted in km. Classify each.

Example 19

medium
Classify the variable 'number of siblings' and give its possible values for a typical respondent.

Example 20

challenge
A variable can take any value in [0,1][0,1]. Argue it has infinitely many possible values, and explain why this forces continuous (not discrete) treatment.

Example 21

challenge
Explain why probability for a continuous variable assigns 00 to any single exact value, unlike a discrete variable.

Example 22

challenge
Classify the set of rational numbers: is it discrete or continuous? Defend carefully.

Example 23

easy
Classify as discrete or continuous: the number of books on a shelf.

Example 24

easy
Classify as discrete or continuous: the volume of milk in a carton.

Example 25

easy
Classify as discrete or continuous: the age of a person reported in years.

Example 26

easy
Classify as discrete or continuous: the length of a fish in centimeters.

Example 27

easy
Discrete or continuous: the air pressure in a tire.

Example 28

medium
A variable takes values in {2,4,6,8,10}\{2, 4, 6, 8, 10\}. Is it discrete or continuous? Justify.

Example 29

medium
A weather station reports temperature rounded to the nearest 0.1∘0.1^\circC. Is the recorded variable discrete or continuous? What about the true temperature?

Example 30

medium
For a discrete random variable, the probabilities of all outcomes must sum to what? What is the analogous statement for a continuous variable?

Example 31

medium
Classify the variable 'time to run a 100 m race in seconds'. Is the timer's recorded result the same as the underlying quantity?

Example 32

medium
Classify each: (a) number of phone calls received per day, (b) battery voltage in volts.

Example 33

medium
True or false: every finite set of numbers is discrete. Justify briefly.

Example 34

medium
A particle's position is given by x=3t2x = 3t^2 for t∈[0,5]t \in [0,5]. Is xx discrete or continuous as a function of tt?

Example 35

medium
A pollster asks 'How many hours did you sleep last night?' and records the answer. Is the underlying quantity discrete or continuous, and what about the recorded response?

Example 36

hard
Explain why we use a probability density function (PDF) rather than a probability mass function (PMF) for continuous variables.

Example 37

hard
Classify each as discrete or continuous: (a) the integers Z\mathbb{Z}, (b) the interval [0,1][0, 1], (c) the set {1/n:n∈Z+}\{1/n : n \in \mathbb{Z}^+\}.

Example 38

hard
A factory produces both rolls of fabric (sold by the meter) and t-shirts (sold per unit). Which quantity is discrete and which continuous, and how does that affect pricing models?

Example 39

hard
Two ways to plot exam scores out of 100: dot plot of integer scores, or histogram with bins of width 10. Which respects discreteness, and what does the histogram trade away?

Example 40

hard
Give a concrete example where treating a discrete variable as continuous gives a wrong probability statement.

Example 41

challenge
The set of irrational numbers in [0,1][0,1] is uncountable. Argue why it is naturally classed as continuous despite missing every rational.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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