Use the definition of continuity and the properties of limits to show that the function

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

If is continuous on , what can you say about its graph?

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

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

Sketch the graph of a function that is continuous everywhere except at x 3 and is continuous from the left at 3.

Sketch the graph of a function that has a jump discontinuity at and a removable discontinuity at , but is continuous elsewhere.

A parking lot charges $3 for the first hour (or part of an hour) and $2 for each succeeding hour (or part), up to a daily maximum of $10. (a) Sketch a graph of the cost of parking at this lot as a function of the time parked there. (b) Discuss the discontinuities of this function and their significance to someone who parks in the lot.

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

If and are continuous functions with and , find .

Use the definition of continuity and the properties of limits to show that the function is continuous at the given number.

Use the definition of continuity and the properties of limits to show that the function is continuous at the given number.

Use the definition of continuity and the properties of limits to show that the function is continuous at the given number.

Use the definition of continuity and the properties of limits to show that the function is continuous on the given interval.

Use the definition of continuity and the properties of limits to show that the function is continuous on the given interval.

Explain why the function is discontinuous at the given number . Sketch the graph of the function.

Explain why the function is discontinuous at the given number . Sketch the graph of the function.

Explain why the function is discontinuous at the given number . Sketch the graph of the function.

Explain why the function is discontinuous at the given number . Sketch the graph of the function.

Explain why the function is discontinuous at the given number . Sketch the graph of the function.

Explain why the function is discontinuous at the given number . Sketch the graph of the function.

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

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

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

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

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

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

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

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

Locate the discontinuities of the function and illustrate by graphing.

Locate the discontinuities of the function and illustrate by graphing.

Use continuity to evaluate the limit.

Use continuity to evaluate the limit.

Use continuity to evaluate the limit.

Use continuity to evaluate the limit.

Show that is continuous on

Show that is continuous on

Find the numbers at which is discontinuous. At which of these numbers is continuous from the right, from the left, or neither? Sketch the graph of .

Find the numbers at which is discontinuous. At which of these numbers is continuous from the right, from the left, or neither? Sketch the graph of .

Find the numbers at which is discontinuous. At which of these numbers is continuous from the right, from the left, or neither? Sketch the graph of .

The gravitational force exerted by Earth on a unit mass at a distance r from the center of the planet is 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 is the function continuous on ?

Find the constant that makes continuous on tx  x 2 c 2 cx  20 if x  4 if x  4 c

Which of the following functions has a removable discontinuity at ? If the discontinuity is removable, find a function that agrees with for and is continuous on R. (a) , (b) , (c) , (d

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

If , show that there is a number such that .

Use the Intermediate Value Theorem to prove that there is a positive number such that . (This proves the existence of the number

Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval.

Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval.

Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval.

Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval.

(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.

(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.

(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.

(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.

Prove that is continuous at if and only if x lim h l 0 f a  h fa

To prove that sine is continuous, we need to show that for every real number . By Exercise 55 an equivalent statement is that 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 is continuous? f x  0 1 if x is rational if x is irrational

For what values of is continuous? tx  0 x if x is rational if x is irrational

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

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

A Tibetan monk leaves the monastery at 7:00 A.M. and takes his usual path to the top of the mountain, arriving at 7:00 P.M. The following morning, he starts at 7:00 A.M. at the top and takes the same path back, arriving at the monastery at 7:00 P.M. 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.