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High School Precalculus: Help and Review32 chapters | 297 lessons

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.

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.

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.

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.

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

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:

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.

It also has to be true that Rome exists. If both roads approach Rome, and Rome has been lifted up into the sky, then it really doesn't matter. You can't get there. If a function is continuous at some point called *a*, *a* must be defined. You might write that this way:

In other words, the function exists at the point *a*, or *a* is in the **domain** of the function.

However, all roads labeled 'To Rome' approaching the same location, and Rome existing aren't quite enough. The place those roads lead to must actually be that same Rome that exists. If the roads all say 'To Rome' and all end up at the same place - Cincinnati, you've got a problem. Rome existing on another continent does you no good.

Similarly, the limit that exists must be equal to the value of the function at that point. Graph C above is an example of this property not being met. Both red sections of the curve lead toward the same spot, but the graph isn't defined there. It is defined further north of that spot - at the blue dot. If the property is satisfied, then mathematically you would write:

A function is continuous in some interval (section of the curve) if you can trace your pencil over the curve in that section without having to lift it. If this is true, then at any point in that interval, the graph will be continuous. It will satisfy the three properties of continuity.

Work to achieve these goals when you conclude this lesson:

- Describe a continuous function
- Graph a continuous function
- Use calculus to determine if a function is continuous

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17 in chapter 19 of the course:

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High School Precalculus: Help and Review32 chapters | 297 lessons

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