Problem 1E: Write an equation that expresses the fact that a function f is continuous at the number 4. Problem 2E: If f is continuous on ( , ).what can you say about its graph? Problem 3E: (a) From the graph of f , state the numbers at which f is discontinuous and explain why. (b) For... Problem 4E: From the graph of g, state the intervals on which g is continuous. Problem 5E: The graph of a function f is given. (a) At what numbers a does limxaf(x) not exist? (b) At what... Problem 6E: The graph of a function f is given. (a) At what numbers a does limxaf(x) not exist? (b) At what... Problem 7E: Sketch the graph of a function f that is defined on and continuous except for the stated... Problem 8E: Sketch the graph of a function f that is defined on and continuous except for the stated... Problem 9E Problem 10E Problem 11E: The toll T charged for driving on a certain stretch of a toll road is 5 except during rush hours... Problem 12E: Explain why each function is continuous or discontinuous. (a) The temperature at a specific location... Problem 13E: Use the definition of continuity and the properties of limits to show that the function is... Problem 14E: Use the definition of continuity and the properties of limits to show that the function is... Problem 15E: Use the definition of continuity and the properties of limits to show that the function is... Problem 16E: Use the definition of continuity and the properties of limits to show that the function is... Problem 17E: Use the definition of continuity and the properties of limits to show that the function is... Problem 18E: Use the definition of continuity and the properties of limits to show that the function is... Problem 19E: Explain why the function is discontinuous at the given number a. Sketch the graph of the function.... Problem 20E: Explain why the function is discontinuous at the given number a. Sketch the graph of the function.... Problem 21E: Explain why the function is discontinuous at the given number a. Sketch the graph of the function.... Problem 22E: Explain why the function is discontinuous at the given number a. Sketch the graph of the function.... Problem 23E: Explain why the function is discontinuous at the given number a. Sketch the graph of the function.... Problem 24E: Explain why the function is discontinuous at the given number a. Sketch the graph of the function.... Problem 25E: (a) Show that f has a removable discontinuity at x=3 . (b) Redefine f(3) so that f is continuous at... Problem 26E: (a) Show that f has a removable discontinuity at x=3 . (b) Redefine f(3) so that f is continuous at... Problem 27E: Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its... Problem 28E: Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its... Problem 29E: Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its... Problem 30E: Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its... Problem 31E Problem 32E: Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its... Problem 33E: Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its... Problem 34E: Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its... Problem 35E: Use continuity to evaluate the limit. limx2x20x2 Problem 36E: Use continuity to evaluate the limit. 36. lim/2sin(tan(cos)) Problem 37E: Use continuity to evaluate the limit. limx1ln(5x21+x) Problem 38E: Use continuity to evaluate the limit. limx43x22x4 Problem 39E: Locate the discontinuities of the function and illustrate by graphing. 39. f(x)=11sinx Problem 40E: Locate the discontinuities of the function and illustrate by graphing. 40. y=arctan1x Problem 41E: Show that f is continuous on ( , ). f(x)={1x2ifx1lnxifx1 Problem 42E: Show that f is continuous on ( , ). f(x)={sinxifx/4cosxifx/4 Problem 43E: Find the numbers at which f is discontinuous. At which of these numbers is f continuous from the... Problem 44E: Find the numbers at which f is discontinuous. At which of these numbers is f continuous from the... Problem 45E: Find the numbers at which f is discontinuous. At which of these numbers is f continuous from the... Problem 46E: The gravitational force exerted by the planet Earth on a unit mass at a distance r from the center... Problem 47E: For what value of the constant c is the function f continuous on ( , )? f(x)={cx2+2xifx2x3cxifx2 Problem 48E: Find the values of a and h that make f continuous everywhere. f(x)={x24x2ifx2ax2bx+3ifxx32xa+bifx2 Problem 49E: Suppose f and g are continuous functions such that g(2) = 6 and limx2[3f(x)+f(x)g(x)]=36. Find f(2). Problem 50E: Let f(x)=1/xandg(x)=1/x2. (a) Find (f g)(x). (b) Is f g continuous everywhere? Explain. Problem 51E: Which of the following functions .f has a removable discontinuity at a? If the discontinuity is... Problem 52E: Suppose that a function f is continuous on [0, 1] except at 0.25 and that f(0) = 1 and f(l) = 3. Let... Problem 53E: If f(x) = x2 + 10 sin x, show that there is a number c such that f(c) = 1000. Problem 54E: Suppose f is continuous on [1, 5] and the only solutions of the equation f(x) = 6 arc x = 1 and x =... Problem 55E: Use the Intermediate Value Theorem to show that there is a solution of the given equation in the... Problem 56E: Use the Intermediate Value Theorem to show that there is a root of the given equation in the... Problem 57E: Use the Intermediate Value Theorem to show that there is a root of the given equation in the... Problem 58E: Use the Intermediate Value Theorem to show that there is a root of the given equation in the... Problem 59E: (a) Prove that the equation has at least one real root. (b) Use your calculator to find an interval... Problem 60E: (a) Prove that the equation has at least one real root. (b) Use your calculator to find an interval... Problem 61E Problem 62E Problem 63E: Prove, without graphing, that the graph of the function has at least two x-intercepts in the... Problem 64E: Prove, without graphing, that the graph of the function has at least two x-intercepts in the... Problem 65E: Prove that f is continuous at a if and only if limh0f(a+h)=f(a) Problem 66E: To prove that sine is continuous, we need to show that limxasinx=sina for every real number a. By... Problem 67E: Prove that cosine is a continuous function. Problem 68E Problem 69E Problem 70E Problem 71E: For what values of x is f continuous? f(x)={0ifxisrational1ifxisirrational Problem 72E: For what values of x is g continuous? g(x)={0ifxisrational1ifxisirrational Problem 73E: Show that the function f(x)={x4sin(1/x)ifx00ifx=0 is continuous on ( , ). Problem 74E: If a and b are positive numbers, prove that the equation ax3+2x21+bx3+x2=0 has at least one solution... Problem 75E: A Tibetan monk leaves the monastery at 7:00 AM and takes his usual path to the top of the mountain,... Problem 76E: Absolute Value and Continuity (a) Show that the absolute value function F(x)=x is continuous... format_list_bulleted
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