Continuous Functions: Properties & Definition

Instructor: Kimberlee Davison

Kim has a Ph.D. in Education and has taught math courses at four colleges, in addition to teaching math to K-12 students in a variety of settings.

In this lesson, you will learn what a continuous function is and how to identify one by tracing your finger or pencil on a graph or by using three properties of continuous functions.


A continuous function is a function, or equation, which is smooth when graphed over the entire domain (set of x-values) that you care about. If you were to run your finger along the graph, you would never need to lift your finger. There are no interruptions or gaps in the graph.

Continuous Functions and the Real World

Imagine that you live next door to the local football stadium and you want to set up a stand to sell snow cones as people arrive for the game. You are trying to figure out where to buy your ice. Green Grocery Store sells ice by the 10 pound bag. Each bag costs you $2.

On the other hand, Red Ice Delivery Service can bring you any amount of ice you want. It costs you $0.25 per pound.

If you buy ice from Green, you will have some waste if you need 15 or 25 pounds because you can only buy ten pound bags, but the cost is a little lower per pound. With Red, you pay a little more per pound but you can get exactly how much you want.

If you look at the graphs of the Red and the Green ice costs, you will see a couple big differences. One difference is that the red graph is a straight line with no breaks in it. The green graph looks like a staircase. There are breaks in the graph at each ten pounds of ice needed. Using Green, if you need 40 pounds of ice, or just under, you can get away with 4 bags of ice - $8 in cost. But as soon as you need a little more than 40, you have to buy one more bag and your cost jumps by $2.

Green Discontinuous Function and Red Continuous Function
Graph showing a continuous and a discontinuous function

Because the green function has breaks, or jumps, in it, it is not continuous. You could not trace a pencil across the entire length of the green graph without having to pick up your pencil. The red function, however, could be drawn without lifting your pencil. It is continuous. There are no sudden jumps in your costs at any point.

Graphs and Continuity

If you look at the four small graphs (Graphs A-D), you will see that only one is continuous (Graph A) over the intervals (x-values) shown. The other three all have breaks in them. You could not draw the entire length of the graph without lifting your pencil.

Graph A
A continuous function

Graph B
A discontinuous function

Graph C
A discontinuous function

Graph D
A discontinuous function

Continuous Functions and Calculus

You can also use calculus to determine whether a function is continuous.

Property 1.

There is a connection between continuous functions and limits, a topic studied in calculus. If a function is continuous at some point, then the limit at that point is the same whether you approach the point from the right or left. In other words, if you move your pencil toward that point from either side, you end up at the same place.

In Graph D above, you end up 'higher' (larger y-value) if you move your pencil toward the '2' along the blue line from the left rather than along the red line from the right. If a is the x-value you are interested in, then mathematically you would write this as:

Limit of f(x) exists as x approaches a

It's a bit like saying that, whether you approach Rome from the North or from the South on a road labeled 'To Rome,' you still are approaching the same place.

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