In exercise, graph the function ƒ to see whether it

PROBLEM 72E Use the Intermediate Value Theorem in exercise to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations.

PROBLEM 4E In Exercise, say whether the function graphed is continuous on If not, where does it fail to be continuous [-1, 3] and why?

PROBLEM 5E Exercises refer to the function graphed in the accompanying figure. The graph for Exercise.

PROBLEM 21E At what points are the functions in exercise continuous?

PROBLEM 19E At what points are the functions in exercise continuous?

PROBLEM 26E At what points are the functions in exercise continuous?

PROBLEM 20E At what points are the functions in exercise continuous?

PROBLEM 28E At what points are the functions in exercise continuous?

PROBLEM 25E At what points are the functions in exercise continuous?

PROBLEM 29E At what points are the functions in exercise continuous?

PROBLEM 27E At what points are the functions in exercise continuous?

PROBLEM 33E Find the limits in exercise. Are the functions continuous at the point being approached?

PROBLEM 31E Find the limits in exercise. Are the functions continuous at the point being approached?

PROBLEM 30E At what points are the functions in exercise continuous?

PROBLEM 32E Find the limits in exercise. Are the functions continuous at the point being approached?

PROBLEM 34E Find the limits in exercise. Are the functions continuous at the point being approached?

PROBLEM 35E Find the limits in exercise. Are the functions continuous at the point being approached?

PROBLEM 36E Find the limits in exercise. Are the functions continuous at the point being approached?

PROBLEM 40E Define h(2) in a way that extends to be continuous at t = 2.

PROBLEM 41E Define ƒ(1) in a way that extends to be continuous at s = 1.

PROBLEM 37E Find the limits in exercise. Are the functions continuous at the point being approached?

PROBLEM 55E Roots of a cubic Show that the equation x3 - 15x + 1 = 0 has three solutions in the interval [-4, 4] .

PROBLEM 56E A function value Show that the function F(x) = (x - a)2 · (x - b)2 + x takes on the value(a + b)/2 for some value of x.

PROBLEM 57E Solving an equation If ƒ(x) = x3 - 8x + 10, show that there are values c for which ƒ(c) equals (c) 5,000,000.

PROBLEM 38E Find the limits in exercise. Are the functions continuous at the point being approached?

PROBLEM 42E Define g(4) in a way that extends to be continuous at x = 4.

PROBLEM 71E Use the Intermediate Value Theorem in exercise to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations.

PROBLEM 78E Use the Intermediate Value Theorem in exercise to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations. (three roots) Make sure you are using radian mode.

PROBLEM 2E In Exercise, say whether the function graphed is continuous on If not, where does it fail to be continuous [-1, 3] and why?

PROBLEM 3E In Exercise, say whether the function graphed is continuous on If not, where does it fail to be continuous [-1, 3] and why?

PROBLEM 1E In Exercise, say whether the function graphed is continuous on If not, where does it fail to be continuous [-1, 3] and why?

PROBLEM 6E Exercises refer to the function graphed in the accompanying figure. The graph for Exercise.

PROBLEM 7E Exercises refer to the function graphed in the accompanying figure. The graph for Exercise. a. Is ƒ defined at x = 2? (Look at the definition of ƒ.) b. Is ƒ continuous x = 2?

PROBLEM 8E Exercises refer to the function graphed in the accompanying figure. The graph for Exercise. At what values of x is ƒ continuous?

PROBLEM 9E Exercises refer to the function graphed in the accompanying figure. The graph for Exercise. What value should be assigned to ƒ(2) to make the extended function continuous at x = 2?

PROBLEM 10E Exercises refer to the function graphed in the accompanying figure. The graph for Exercise. To what new value should ƒ(1) be changed to remove the discontinuity?

PROBLEM 11E At which points do the functions in exercise fail to be continuous? At which points, if any, are the discontinuities removable? Not removable? Give reasons for your answers. Exercise 1, Section 2.4 Which of the following statements about the function y = ƒ(x) graphed here are true, and which are false?

PROBLEM 13E At what points are the functions in exercise continuous?

PROBLEM 12E At which points do the functions in exercise fail to be continuous? At which points, if any, are the discontinuities removable? Not removable? Give reasons for your answers. Exercise 2, Section 2.4 Which of the following statements about the function y = ƒ(x) graphed here are true, and which are false?

PROBLEM 14E At what points are the functions in exercise continuous?

PROBLEM 15E At what points are the functions in exercise continuous?

PROBLEM 16E At what points are the functions in exercise continuous?

PROBLEM 17E At what points are the functions in exercise continuous?

PROBLEM 18E At what points are the functions in exercise continuous?

PROBLEM 22E At what points are the functions in exercise continuous?

PROBLEM 23E At what points are the functions in exercise continuous?

PROBLEM 24E At what points are the functions in exercise continuous?

PROBLEM 39E Define g(3) in a way that extends to be continuous at x = 3.

PROBLEM 43E For what value of a is continuous at every x?

PROBLEM 44E For what value of b is continuous at every x?

PROBLEM 45E For what values of a is continuous at every x?

PROBLEM 46E For what value of b is continuous at every x?

PROBLEM 47E For what values of a and b is continuous at every x?

PROBLEM 49E In exercise, graph the function ƒ to see whether it appears to have a continuous extension to the origin. If it does, use Trace and Zoom to find a good candidate for the extended function’s value at x = 0.If the function does not appear to have a continuous extension, can it be extended to be continuous at the origin from the right or from the left? If so, what do you think the extended function’s value(s) should be? .

PROBLEM 51E In exercise, graph the function ƒ to see whether it appears to have a continuous extension to the origin. If it does, use Trace and Zoom to find a good candidate for the extended function’s value at x = 0.If the function does not appear to have a continuous extension, can it be extended to be continuous at the origin from the right or from the left? If so, what do you think the extended function’s value(s) should be?

PROBLEM 50E In exercise, graph the function ƒ to see whether it appears to have a continuous extension to the origin. If it does, use Trace and Zoom to find a good candidate for the extended function’s value at x = 0.If the function does not appear to have a continuous extension, can it be extended to be continuous at the origin from the right or from the left? If so, what do you think the extended function’s value(s) should be?

PROBLEM 48E For what values of a and b is continuous at every x?

PROBLEM 52E In exercise, graph the function ƒ to see whether it appears to have a continuous extension to the origin. If it does, use Trace and Zoom to find a good candidate for the extended function’s value at x = 0.If the function does not appear to have a continuous extension, can it be extended to be continuous at the origin from the right or from the left? If so, what do you think the extended function’s value(s) should be?

PROBLEM 53E A continuous function y = ƒ(x) is known to be negative at x = 0 and positive at x = 1.Why does the equation ƒ(x) = 0 have at least one solution between x = 0 and x = 1? Illustrate with a sketch.

PROBLEM 54E Explain why the equation cos x = x has at least one solution.

PROBLEM 59E Removable discontinuity Give an example of a function ƒ(x) that is continuous for all values of x except x = 2, where it has a removable discontinuity. Explain how you know that ƒ is discontinuous at x = 2, and how you know the discontinuity is removable.

PROBLEM 58E Explain why the following five statements ask for the same information. a. Find the roots of ƒ(x) = x3 - 3x - 1. b. Find the x-coordinates of the points where the curve y = x3 crosses the line y = 3x + 1. c. Find all the values of x for which x3 - 3x = 1. d. Find the x-coordinates of the points where the cubic curve y = x3 - 3x crosses the line y = 1 e. Solve the equation x3 - 3x - 1 = 0.

PROBLEM 60E Nonremovable discontinuity Give an example of a function g(x) that is continuous for all values of x except x = -1, where it has a nonremovable discontinuity. Explain how you know that g is discontinuous there and why the discontinuity is not removable.

PROBLEM 61E A function discontinuous at every point a. Use the fact that every nonempty interval of real numbers contains both rational and irrational numbers to show that the function is discontinuous at every point. b. Is ƒ right-continuous or left-continuous at any point?

PROBLEM 62E If functions ƒ(x) and g(x) are continuous for 0 ? x ? 1, could ƒ(x)/g(x) possibly be discontinuous at a point of [0, 1]? Give reasons for your answer.

PROBLEM 63E If the product function is continuous x = 0, at must ƒ(x) and g(x) be continuous x = 0, at Give reasons for your answer

PROBLEM 64E Discontinuous composite of continuous functions Give an example of functions ƒ and g, both continuous at x = 0, for which the composite is discontinuous at x = 0. Does this contradict Theorem 9? Give reasons for your answer.

PROBLEM 65E Never-zero continuous functions Is it true that a continuous function that is never zero on an interval never changes sign on that interval? Give reasons for your answer.

PROBLEM 67E A fixed point theorem Suppose that a function ƒ is continuous on the closed interval [0, 1] and that 0 ? f(x) ? 1for every x in [0, 1]. Show that there must exist a number c in [0, 1] such that ƒ(c) = c (c is called a fixed point of ƒ).

PROBLEM 66E Stretching a rubber band Is it true that if you stretch a rubber band by moving one end to the right and the other to the left, some point of the band will end up in its original position? Give reasons for your answer.

PROBLEM 68E The sign-preserving property of continuous functions Let ƒ be defined on an interval (a, b) and suppose that ƒ(c) ? 0 at some c where ƒ is continuous. Show that there is an interval about c where ƒ has the same sign as ƒ(c).

PROBLEM 69E Prove that ƒ is continuous at c if and only if

PROBLEM 70E Use Exercise 69 together with the identities to prove that both f(x) = sin x and g(x) = cos x are continuous at every point x = c. Exercise 69:

PROBLEM 73E Use the Intermediate Value Theorem in exercise to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations.

PROBLEM 74E Use the Intermediate Value Theorem in exercise to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations.

PROBLEM 75E Use the Intermediate Value Theorem in exercise to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations.

PROBLEM 76E Use the Intermediate Value Theorem in exercise to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations.

PROBLEM 77E Use the Intermediate Value Theorem in exercise to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations. (one root) Make sure you are using radian mode.