(a) Show that the absolute value function Fsxd

Write an equation that expresses the fact that a function f is continuous at the number 4.

. If f is continuous on s2`, `d, what can you say about its graph?

(a) From the graph of f, state the numbers at which f is discontinuous and explain why. (b) For each of the numbers stated in part (a), determine whether f is continuous from the right, or from the left, or neither.

From the graph of t, state the intervals on which t is continuous.

Sketch the graph of a function f that is continuous except for the stated discontinuity Discontinuous, but continuous from the right, at 2

Sketch the graph of a function f that is continuous except for the stated discontinuity Discontinuities at 21 and 4, but continuous from the left at 21 and from the right at 4

Sketch the graph of a function f that is continuous except for the stated discontinuity Removable discontinuity at 3, jump discontinuity at 5

Sketch the graph of a function f that is continuous except for the stated discontinuity Neither left nor right continuous at 22, continuous only from the left at 2

The toll T charged for driving on a certain stretch of a toll road is $5 except during rush hours (between 7 am and 10 am and between 4 pm and 7 pm) when the toll is $7. (a) Sketch a graph of T as a function of the time t, measured in hours past midnight. (b) Discuss the discontinuities of this function and their significance to someone who uses the road.

Explain why each function is continuous or discontinuous. (a) The temperature at a specific location as a function of time (b) The temperature at a specific time as a function of the distance due west from New York City (c) The altitude above sea level as a function of the distance due west from New York City (d) The cost of a taxi ride as a function of the distance traveled (e) The current in the circuit for the lights in a room as a function of time

Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a. fsxd sx 1 2x 3 d 4 , a 21

Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a. tstd t 2 1 5t 2t 1 1 , a 2

Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a. psvd 2s3v2 1 1 , a 1

Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a.fsxd 3x4 2 5x 1 s 3 x 2 1 4 , a 2

Use the definition of continuity and the properties of limits to show that the function is continuous on the given interval. fsxd x 1 sx 2 4 , f4, `d

Use the definition of continuity and the properties of limits to show that the function is continuous on the given interval. tsxd x 2 1 3x 1 6 , s2`, 22d

Explain why the function is discontinuous at the given number a. Sketch the graph of the function. fsxd 1 x 1 2 a 22

Explain why the function is discontinuous at the given number a. Sketch the graph of the function. fsxd H 1 x 1 2 1 if x 22 if x 22 a 22

Explain why the function is discontinuous at the given number a. Sketch the graph of the function. fsxd H x 1 3 2x if x < 21 if x . 21 a 21

Explain why the function is discontinuous at the given number a. Sketch the graph of the function. fsxd H x 2 2 x x 2 2 1 1 if x 1 if x 1 a 1

Explain why the function is discontinuous at the given number a. Sketch the graph of the function. fsxd H cos x 0 1 2 x 2 if x , 0 if x 0 if x . 0 a 0

Explain why the function is discontinuous at the given number a. Sketch the graph of the function.fsxd H 2x 2 2 5x 2 3 x 2 3 6 if x 3 if x 3 a 3

How would you remove the discontinuity of f ? In other words, how would you define fs2d in order to make f continuous at 2? fsxd x 2 2 x 2 2 x 2 2

How would you remove the discontinuity of f ? In other words, how would you define fs2d in order to make f continuous at 2? fsxd x 3 2 8 x 2 2 4

Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. Fsxd 2x 2 2 x 2 1 x 2 1 1

Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. Gsxd x 2 1 1 2x 2 2 x 2 1

Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. Gsxd x 2 1 1 2x 2 2 x 2 1

Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. Rstd esin t 2 1 cos t

Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. Astd arcsins1 1 2td

Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. Bsxd tan x s4 2 x 2

Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain.Msxd 1 1 1 x

Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. Nsrd tan21 s1 1 e2r 2 d

Locate the discontinuities of the function and illustrate by graphing. y 1 1 1 e 1yx

Locate the discontinuities of the function and illustrate by graphing. y lnstan2 x

Use continuity to evaluate the limit. lim xl2 x s20 2 x 2

Use continuity to evaluate the limit. lim xl sinsx 1 sin xd

Use continuity to evaluate the limit. lim xl1 lnS 5 2 x 2 1 1 x D

Use continuity to evaluate the limit. lim xl4 3sx 222x24

Show that f is continuous on s2`, `d. fsxd H 1 2 x 2 if x < 1 lnx if x . 1

Show that f is continuous on s2`, `d. fsxd H sin x if x , y4 cos x if x > y4

Find the numbers at which f is discontinuous. At which of these numbers is f continuous from the right, from the left, or neither? Sketch the graph of f. fsxd H x 2 x 1yx if x , 21 if 21 < x , 1 if x > 1

Find the numbers at which f is discontinuous. At which of these numbers is f continuous from the right, from the left, or neither? Sketch the graph of f. fsxd H 2x 3 2 x sx if x < 1 if 1 , x < 4 if x . 4

Find the numbers at which f is discontinuous. At which of these numbers is f continuous from the right, from the left, or neither? Sketch the graph of f. fsxd H x 1 2 ex 2 2 x if x , 0 if 0 < x < 1 if x . 1

The gravitational force exerted by the planet Earth on a unit mass at a distance r from the center of the planet is Fsrd GMr R3 if r , R GM r 2 if r > R where M is the mass of Earth, R is its radius, and G is the gravitational constant. Is F a continuous function of r?

For what value of the constant c is the function f continuous on s2`, `d? fsxd H cx 2 1 2x x 3 2 cx if x , 2 if x > 2

Find the values of a and b that make f continuous everywhere. fsxd x 2 2 4 x 2 2 ax 2 2 bx 1 3 2x 2 a 1 b if x , 2 if 2 < x , 3 if x > 3

Suppose f and t are continuous functions such that ts2d 6 and limx l2 f3fsxd 1 fsxd tsxdg 36. Find fs2d.

Let fsxd 1yx and tsxd 1yx 2 . (a) Find s f + tds xd. (b) Is f + t continuous everywhere? Explain

Which of the following functions f has a removable discontinuity at a? If the discontinuity is removable, find a function t that agrees with f for x a and is continuous at a. (a) fsxd x 4 2 1 x 2 1 , a 1 (b) fsxd x 3 2 x 2 2 2x x 2 2 , a 2 (c) fsxd v sin x b, a

Suppose that a function f is continuous on [0, 1] except at 0.25 and that fs0d 1 and fs1d 3. Let N 2. Sketch two possible graphs of f, one showing that f might not satisfy the conclusion of the Intermediate Value Theorem and one showing that f might still satisfy the conclusion of the Intermediate Value Theorem (even though it doesnt satisfy the hypothesis).

If fsxd x 2 1 10 sin x, show that there is a number c such that fscd 1000

Suppose f is continuous on f1, 5g and the only solutions of the equation fsxd 6 are x 1 and x 4. If fs2d 8, explain why fs3d . 6.

Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. x 4 1 x 2 3 0, s1, 2d

Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. ln x x 2 sx , s2, 3d

Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. ex 3 2 2x, s0, 1d

Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval.sin x x 2 2 x, s1, 2d

8 (a) Prove that the equation has at least one real root. (b) Use your calculator to find an interval of length 0.01 that contains a root cos x x 3

8 (a) Prove that the equation has at least one real root. (b) Use your calculator to find an interval of length 0.01 that contains a root ln x 3 2 2x

(a) Prove that the equation has at least one real root. (b) Use your graphing device to find the root correct to three decimal places. 100e2xy100 0.01x 2

(a) Prove that the equation has at least one real root. (b) Use your graphing device to find the root correct to three decimal places. arctan x 1 2 x

Prove, without graphing, that the graph of the function has at least two x-intercepts in the specified interval. y sin x 3 , s1, 2d

Prove, without graphing, that the graph of the function has at least two x-intercepts in the specified interval. . y x 2 2 3 1 1yx, s0, 2d

Prove that f is continuous at a if and only if lim h l0 fsa 1 hd fsa

To prove that sine is continuous, we need to show that limx l a sin x sin a for every real number a. By Exercise 63 an equivalent statement is that lim h l0 sinsa 1 hd sin a Use (6) to show that this is true.

Prove that cosine is a continuous function.

(a) Prove Theorem 4, part 3. (b) Prove Theorem 4, part 5.

For what values of x is f continuous? fsxd H 0 1 if x is rational if x is irrational

For what values of x is t continuous? tsxd H 0 x if x is rational if x is irrational

Is there a number t hat is exactly 1 more than its cube?

Is there a number t hat is exactly 1 more than its cube?

Show that the function fsxd H x 4 sins1yxd 0 if x 0 if x 0 is continuous on s2`, `d.

(a) Show that the absolute value function Fsxd | x | is continuous everywhere. (b) Prove that if f is a continuous function on an interval, then so is | f |. (c) Is the converse of the statement in part (b) also true? In other words, if | f | is continuous, does it follow that f is continuous? If so, prove it. If not, find a counterexample.

A Tibetan monk leaves the monastery at 7:00 am and takes his usual path to the top of the mountain, arriving at 7:00 pm. The following morning, he starts at 7:00 am at the top and takes the same path back, arriving at the monastery at 7:00 pm. Use the Intermediate Value Theorem to show that there is a point on the path that the monk will cross at exactly the same time of day on both days.