Continuous functions have important properties. For example, if a function is continuous on a closed interval, it attains a maximum value and a minimum value on that interval. This property is very useful when dealing with optimization problems. Continuous functions have the intermediate value property; that is, whenever they take on two values, they also take on all values in between. One immediate application of the intermediate value property is an approximate method of finding roots called the bisection method. Also we will learn later that continuous functions are integrable.
The following theorem states an important property of continuous functions.
Theorem (Bolzano's Theorem): If is continuous on and if and have different signs, then there exists a point in such that .
- In other words, if is a continuous function on and if and (or conversely if and ), then takes on 0 at least once in that interval. Equivalently, we can say that has a root (or zero) in that interval.
Geometrically this theorem is intuitive because it merely tells us that the curve of a continuous function, which begins below the -axis and ends above it, must intersect the -axis at some point in between (see Figure 1).
Figure 1 If is continuous and then the graph of cuts the -axis somewhere between and .
- Bolzano's Theorem guarantees the existence of one zero (or root), but the equation may have more than one solution (see Figure 2).
Figure 2 If is continuous and , Bolzano's theorem assures us that there is at least one solution for the equation between and , but there may be more than one solution as we see in this figure.
- It is possible that (that is, the signs of and are the same) but has a solution in (see Figure 3)
Figure 3 Even if , the equation of may have a solution.
- Note that if is discontinuous even at one point in , the theorem may not hold anymore. For example, consider the function . Here and is continuous everywhere except at . Here Bolzano's Theorem does not hold and the graph of does not intersect the -axis between 1 and 3 (see Figure 4).
Figure 4 Graph of . Here Bolzano's theorem does not apply because is discontinuous at .
- As we can see from Figure 5, if , the continuity of on the open interval is not enough to assure us that has a solution between and . To apply Bolzano's Theorem, the left-continuity at and the right-continuity at are also required.
Figure 5 Here although is continuous on the open interval , because it is not left-continuous at and right-continuous at , Bolzano's theorem does not apply.
A slight generalization of Bolzano's theorem is called the Intermediate Value Theorem:
Theorem (Intermediate Value Theorem): Let be a continuous function on the closed interval . If is a number between and , then there exists a point in such that .
- In other words, takes on any given value between and . The graph of between and is unbroken and any horizontal line between and intersects the graph of at least once.
Let consider the function defined by Because is a continuous function on and have different signs at the two ends of the interval, it follows from Bolzano's theorem that there is a point in such that or .
- As an example of the application of the Intermediate Value Theorem, consider a moving vehicle. If the speedometer shows 100 kilometer per hour, then for any speed between 0 and 100 km/hr, there must be a time when the speed of the car was exactly . If you are 5 feet 8 inches tall, there must be a time when you were exactly 5 feet 2.5 inches.
Use Bolzano's Theorem to show that the function has a zero in the interval . is a polynomial, so it is continuous everywhere including on the interval . On the other hand, Because and have opposite signs, the conditions of Bolzano's Theorem are satisfied and we conclude that has a zero in as shown in Figure. 6
Figure 6 Graph of
Show that if is continuous on and if for every in , then there is at least a point () such that . If or then or . If and , we introduce a new function We have because and and because and . Since is a continuous function on the closed interval and and have opposite signs, it follows from Bolzano's Theorem that there is a point between 0 and 1 such that or .
The geometric interpretation is simple. If and , then the graph of has to cut the line at some point between 0 and 1 (see the following figures).
Let be a function defined on a set . We say has an absolute maximum on (or in) , if there is at least one point in such that for every in . In this case, we say is the point of absolute maximum and is the absolute maximum value (or simply maximum) of on .
Similarly we say has an absolute minimum on , if there exists a point in such that for all in . In this case, we say is the point of absolute minimum and is the minimum value (or simply minimum) of on .
The term absolute extremum refers to either absolute maximum or absolute minimum.
For example, consider the function defined by The absolute maximum of occurs at and its absolute minimum occurs at . The maximum value of is and its absolute minimum is . The graph of is sketched in the following figure.
Figure 7 Graph of . The maximum value of is and the minimum value of is .
Theorem (Extreme Value Theorem) If is continuous on a closed interval , then attains both an absolute maximum and an absolute minimum in
- The above theorem states that if is continuous on , then there are numbers and in such that for all in
- It follows from the above theorem that if is a continuous function on then is bounded on . Let and , so for all in
Although the above theorem is intuitively plausible, a proof of this theorem is not within the scope of an elementary course.
The extreme value theorem states two conditions together are sufficient to ensure that a function has both a minimum and a maximum value on an interval:
- is a closed interval.
- is continuous at the entire points of (for the end points, we need left-continuity or right-continuity).
If any of these two conditions fails, the theorem may not hold anymore.
If is continuous on , then whether or not it may have absolute extremum
The above theorem guarantees the existence of extreme-values, but it does not tell us anything about how to find them. Later on we will develop some tools to help us find the extreme values of functions.