2.2 Continuity

1. If and are continuous functions with and , find .

http://matrix.skku.ac.kr/cal-lab/cal-2-2-1.html

2. If and are continuous functions with and , find .

Since is continuous, .

3. Show that the function is discontinuous at .

http://matrix.skku.ac.kr/cal-lab/cal-2-2-3.html

We may draw with the following Sage command. It shows the function diverge

(+infinity) at . This shows f(x) is discontinuous at .

If we use the following Sage command, we may get limit(abs(ln((x-2)^2))) at directly.

It shows is discontinuous at .

+Infinity

4-7. Determine the points of discontinuity of . At which of these numbers is continuous from the right, from the left or neither? Sketch the graph of .

4.

http://matrix.skku.ac.kr/cal-lab/cal-2-2-4.html

We may draw with the following Sage command.

It shows the function is discontinuous at .

5.

We see that exists for all except . Notice that the right and left limits are different .

6.

We see that exists for all a except . Notice that the right and left limits are different and we see that exists for all a except . Notice that the right and left limits are different .

7.

We see that exists for all except . Notice that the right and left limits are different and we see that exists for all except . Notice that the right and left limits are different .

8-10. For what values of the constant is the function continuous on ?

8.

Thus, for to be continuous on

9.

http://matrix.skku.ac.kr/cal-lab/cal-2-2-9.html

10.

http://matrix.skku.ac.kr/cal-lab/cal-2-2-10.html

Since is not continuous at , and , the solution is .

11-13. Show that the following functions has the removable discontinuity at . Also find a function that agrees with for and is continuous on ?.

11.

for . The discontinuity is removable and agrees with for and is continuous on ?.

12.

for . The discontinuity is removable and agrees with for and is continuous on ?.

13.

for . The discontinuity is removable and agrees with for and is continuous on ?.

14. Let . Is removable discontinuity?

Since ,

is not a removable discontinuity.

15. If , show that there is a number such that .

is continuous on the interval , and . Since , there is a number in such that by the Intermediate Value Theorem.

16. Prove using Intermediate Value Theorem that there is a positive number such that .

Let . We know that is continuous on the interval , and . Since , there is a number in such that by the Intermediate Value Theorem.

17-22. Prove that there is a root of the given equation in the specified interval by using the Intermediate Value Theorem.

17.

http://matrix.skku.ac.kr/cal-lab/cal-2-2-17.html

18. ,

http://matrix.skku.ac.kr/cal-lab/cal-2-2-18.html

True

True

19.

http://matrix.skku.ac.kr/cal-lab/cal-2-2-19.html

-1

1

20. ,

http://matrix.skku.ac.kr/cal-lab/cal-2-2-20.html

True

True

21.

is continuous on the interval , and .

Since , there is a number in such that by the Intermediate Value Theorem.

Thus, there is root of the equation in the interval .

22.

is continuous on the interval , and . Since –6.696<0<e, there is a number in such that by the Intermediate Value Theorem. Thus, there is root of the equation in the interval .

23-26. Show that each of the following equations has at least one real root.

23.

http://matrix.skku.ac.kr/cal-lab/cal-2-2-23.html

http://math2.skku.ac.kr/home/pub/37

We may draw both and in one graph to find intersections. It shows the function has two real roots in and .

The following Sage commands give the value of the intersection in each interval, or .

0.37055809596982464

1.3649584337330951

24.

Let . Then and . So by the Intermediate Value Theorem.

There is a number in (-1,0) such that . This implies that .

25.

Let . Then and . So by the Intermediate Value Theorem.

There is a number in (1,2) such that . This implies that

26.

Let . Then and , and is continuous

So by the Intermediate Value Theorem, there is a number in such that .

This implies that

27-28. Find the values of for which is continuous.

27.

is not continuous at each points. So such value of don't exist.

28.

is continuous at .