Stewart Section 2.5: Continuity

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Stewart Section 2.5: Continuity 0/73
1 Limits and Continuity · Level 2
Write an equation that expresses the fact that a function \(f\) is continuous at the number 4.
2 Limits and Continuity · Level 2
If \(f\) is continuous on \((-\infty, \infty)\), what can you say about its graph?
3 Limits and Continuity · Level 3
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(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.

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4 Limits and Continuity · Level 3
From the graph of \(g\), state the intervals on which \(g\) is continuous.
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5 Limits and Continuity · Level 3
Sketch the graph of a function \(f\) that is continuous except for the stated discontinuity. Discontinuous, but continuous from the right, at 2
6 Limits and Continuity · Level 3
Sketch the graph of a function \(f\) that is continuous except for the stated discontinuity. Discontinuities at \(-1\) and 4, but continuous from the left at \(-1\) and from the right at 4
7 Limits and Continuity · Level 3
Sketch the graph of a function \(f\) that is continuous except for the stated discontinuity. Removable discontinuity at 3, jump discontinuity at 5
8 Limits and Continuity · Level 3
Sketch the graph of a function \(f\) that is continuous except for the stated discontinuity. Neither left nor right continuous at \(-2\), continuous only from the left at 2
9 Limits and Continuity · Level 3
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.

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10 Limits and Continuity · Level 3
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

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11 Limits and Continuity · Level 3
Use the definition of continuity and the properties of limits to show that the function is continuous at the given number \(a\). \(f(x) = (x + 2x^3)^4\), \(a = -1\)
12 Limits and Continuity · Level 3
Use the definition of continuity and the properties of limits to show that the function is continuous at the given number \(a\). \(g(t) = \dfrac{t^2 + 5t}{2t + 1}\), \(a = 2\)
13 Limits and Continuity · Level 3
Use the definition of continuity and the properties of limits to show that the function is continuous at the given number \(a\). \(p(v) = 2\sqrt{3v^2 + 1}\), \(a = 1\)
14 Limits and Continuity · Level 3
Use the definition of continuity and the properties of limits to show that the function is continuous at the given number \(a\). \(f(x) = 3x^4 - 5x + \sqrt[3]{x^2 + 4}\), \(a = 2\)
15 Limits and Continuity · Level 3
Use the definition of continuity and the properties of limits to show that the function is continuous on the given interval. \(f(x) = x + \sqrt{x - 4}\), \([4, \infty)\)
16 Limits and Continuity · Level 3
Use the definition of continuity and the properties of limits to show that the function is continuous on the given interval. \(g(x) = \dfrac{x - 1}{3x + 6}\), \((-\infty, -2)\)
17 Limits and Continuity · Level 3
Explain why the function is discontinuous at the given number \(a\). Sketch the graph of the function. \(f(x) = \dfrac{1}{x + 2}\), \(a = -2\)
18 Limits and Continuity · Level 3
Explain why the function is discontinuous at the given number \(a\). Sketch the graph of the function. \(f(x) = \begin{cases} \dfrac{1}{x + 2} & \quad \text{if } x \neq -2 \\ 1 & \quad \text{if } x = -2 \end{cases}\), \(a = -2\)
19 Limits and Continuity · Level 3
Explain why the function is discontinuous at the given number \(a\). Sketch the graph of the function. \(f(x) = \begin{cases} x + 3 & \quad \text{if } x \leq -1 \\ 2^x & \quad \text{if } x > -1 \end{cases}\), \(a = -1\)
20 Limits and Continuity · Level 3
Explain why the function is discontinuous at the given number \(a\). Sketch the graph of the function. \(f(x) = \begin{cases} \dfrac{x^2 - x}{x^2 - 1} & \quad \text{if } x \neq 1 \\ 1 & \quad \text{if } x = 1 \end{cases}\), \(a = 1\)
21 Limits and Continuity · Level 3
Explain why the function is discontinuous at the given number \(a\). Sketch the graph of the function. \(f(x) = \begin{cases} \cos x & \quad \text{if } x < 0 \\ 0 & \quad \text{if } x = 0 \\ 1 - x^2 & \quad \text{if } x > 0 \end{cases}\), \(a = 0\)
22 Limits and Continuity · Level 3
Explain why the function is discontinuous at the given number \(a\). Sketch the graph of the function. \(f(x) = \begin{cases} \dfrac{2x^2 - 5x - 3}{x - 3} & \quad \text{if } x \neq 3 \\ 6 & \quad \text{if } x = 3 \end{cases}\), \(a = 3\)
23 Limits and Continuity · Level 3
How would you "remove the discontinuity" of \(f\)? In other words, how would you define \(f(2)\) in order to make \(f\) continuous at 2? \(f(x) = \dfrac{x^2 - x - 2}{x - 2}\)
24 Limits and Continuity · Level 3
How would you "remove the discontinuity" of \(f\)? In other words, how would you define \(f(2)\) in order to make \(f\) continuous at 2? \(f(x) = \dfrac{x^3 - 8}{x^2 - 4}\)
25 Limits and Continuity · Level 3
Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. \(F(x) = \dfrac{2x^2 - x - 1}{x^2 + 1}\)
26 Limits and Continuity · Level 3
Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. \(G(x) = \dfrac{x^2 + 1}{2x^2 - x - 1}\)
27 Limits and Continuity · Level 3
Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. \(Q(x) = \dfrac{\sqrt[3]{x - 2}}{x^3 - 2}\)
28 Limits and Continuity · Level 3
Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. \(R(t) = \dfrac{e^{\sin t}}{2 + \cos \pi t}\)
29 Limits and Continuity · Level 3
Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. \(A(t) = \arcsin(1 + 2t)\)
30 Limits and Continuity · Level 3
Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. \(B(x) = \dfrac{\tan x}{\sqrt{4 - x^2}}\)
31 Limits and Continuity · Level 3
Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. \(M(x) = \sqrt{1 + \dfrac{1}{x}}\)
32 Limits and Continuity · Level 3
Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. \(N(r) = \arctan(1 + e^{-r^2})\)
33 Limits and Continuity · Level 4
Locate the discontinuities of the function and illustrate by graphing. \(y = \dfrac{1}{1 + e^{\dfrac{1}{x}}}\)
34 Limits and Continuity · Level 4
Locate the discontinuities of the function and illustrate by graphing. \(y = \ln(\tan^2 x)\)
35 Limits and Continuity · Level 3
Use continuity to evaluate the limit. \(\operatorname*{lim}\limits_{x \rightarrow 2} x \sqrt{20 - x^2}\)
36 Limits and Continuity · Level 3
Use continuity to evaluate the limit. \(\operatorname*{lim}\limits_{x \rightarrow \pi} \sin(x + \sin x)\)
37 Limits and Continuity · Level 3
Use continuity to evaluate the limit. \(\operatorname*{lim}\limits_{x \rightarrow 1} \ln\left(\dfrac{5 - x^2}{1 + x}\right)\)
38 Limits and Continuity · Level 3
Use continuity to evaluate the limit. \(\operatorname*{lim}\limits_{x \rightarrow 4} 3^{\sqrt{x^2 - 2x - 4}}\)
39 Limits and Continuity · Level 4
Show that \(f\) is continuous on \((-\infty, \infty)\). \(f(x) = \begin{cases} 1 - x^2 & \quad \text{if } x \leq 1 \\ \ln x & \quad \text{if } x > 1 \end{cases}\)
40 Limits and Continuity · Level 4
Show that \(f\) is continuous on \((-\infty, \infty)\). \(f(x) = \begin{cases} \sin x & \quad \text{if } x < \dfrac{\pi}{4} \\ \cos x & \quad \text{if } x \geq \dfrac{\pi}{4} \end{cases}\)
41 Limits and Continuity · Level 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 \(f\). \(f(x) = \begin{cases} x^2 & \quad \text{if } x < -1 \\ x & \quad \text{if } -1 \leq x < 1 \\ \dfrac{1}{x} & \quad \text{if } x \geq 1 \end{cases}\)
42 Limits and Continuity · Level 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 \(f\). \(f(x) = \begin{cases} 2^x & \quad \text{if } x \leq 1 \\ 3 - x & \quad \text{if } 1 < x \leq 4 \\ \sqrt{x} & \quad \text{if } x > 4 \end{cases}\)
43 Limits and Continuity · Level 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 \(f\). \(f(x) = \begin{cases} x + 2 & \quad \text{if } x < 0 \\ e^x & \quad \text{if } 0 \leq x \leq 1 \\ 2 - x & \quad \text{if } x > 1 \end{cases}\)
44 Limits and Continuity · Level 4
The gravitational force exerted by the planet Earth on a unit mass at a distance \(r\) from the center of the planet is \(F(r) = \begin{cases} \dfrac{G M r}{R^3} & \quad \text{if } r < R \\ \dfrac{G M}{r^2} & \quad \text{if } r \geq R \end{cases}\) where \(M\) is the mass of Earth, \(R\) is its radius, and \(G\) is the gravitational constant. Is \(F\) a continuous function of \(r\)?
45 Limits and Continuity · Level 4
For what value of the constant \(c\) is the function \(f\) continuous on \((-\infty, \infty)\)? \(f(x) = \begin{cases} c x^2 + 2x & \quad \text{if } x < 2 \\ x^3 - c x & \quad \text{if } x \geq 2 \end{cases}\)
46 Limits and Continuity · Level 5
Find the values of \(a\) and \(b\) that make \(f\) continuous everywhere. \(f(x) = \begin{cases} \dfrac{x^2 - 4}{x - 2} & \quad \text{if } x < 2 \\ a x^2 - b x + 3 & \quad \text{if } 2 \leq x < 3 \\ 2x - a + b & \quad \text{if } x \geq 3 \end{cases}\)
47 Limits and Continuity · Level 4
Suppose \(f\) and \(g\) are continuous functions such that \(g(2) = 6\) and \(\operatorname*{lim}\limits_{x \rightarrow 2} [3f(x) + f(x) g(x)] = 36\). Find \(f(2)\).
48 Limits and Continuity · Level 4
Let \(f(x) = \dfrac{1}{x}\) and \(g(x) = 1/x^2\).
(a) Find \((f \circ g)(x)\).
(b) Is \(f \circ g\) continuous everywhere? Explain.

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49 Limits and Continuity · Level 4
Which of the following functions \(f\) has a removable discontinuity at \(a\)? If the discontinuity is removable, find a function \(g\) that agrees with \(f\) for \(x \neq a\) and is continuous at \(a\).
(a) \(f(x) = \dfrac{x^4 - 1}{x - 1}\), \(a = 1\)
(b) \(f(x) = \dfrac{x^3 - x^2 - 2x}{x - 2}\), \(a = 2\)
(c) \(f(x) = \lfloor \sin x \rfloor\), \(a = \pi\)

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50 Limits and Continuity · Level 4
Suppose that a function \(f\) is continuous on \([0, 1]\) except at 0.25 and that \(f(0) = 1\) and \(f(1) = 3\). 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 doesn't satisfy the hypothesis).
51 Limits and Continuity · Level 3
If \(f(x) = x^2 + 10 \sin x\), show that there is a number \(c\) such that \(f(c) = 1000\).
52 Limits and Continuity · Level 4
Suppose \(f\) is continuous on \([1, 5]\) and the only solutions of the equation \(f(x) = 6\) are \(x = 1\) and \(x = 4\). If \(f(2) = 8\), explain why \(f(3) > 6\).
53 Limits and Continuity · Level 3
Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. \(x^4 + x - 3 = 0\), \((1, 2)\)
54 Limits and Continuity · Level 3
Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. \(\ln x = x - \sqrt{x}\), \((2, 3)\)
55 Limits and Continuity · Level 3
Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. \(e^x = 3 - 2x\), \((0, 1)\)
56 Limits and Continuity · Level 3
Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. \(\sin x = x^2 - x\), \((1, 2)\)
57 Limits and Continuity · Level 4
(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. \(\cos x = x^3\)

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58 Limits and Continuity · Level 4
(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. \(\ln x = 3 - 2x\)

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59 Limits and Continuity · Level 4
(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. \(100 e^{-\dfrac{x}{100}} = 0.01 x^2\)

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60 Limits and Continuity · Level 4
(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. \(\arctan x = 1 - x\)

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61 Limits and Continuity · Level 4
Prove, without graphing, that the graph of the function has at least two \(x\)-intercepts in the specified interval. \(y = \sin x^3\), \((1, 2)\)
62 Limits and Continuity · Level 4
Prove, without graphing, that the graph of the function has at least two \(x\)-intercepts in the specified interval. \(y = x^2 - 3 + \dfrac{1}{x}\), \((0, 2)\)
63 Limits and Continuity · Level 5
Prove that \(f\) is continuous at \(a\) if and only if \(\operatorname*{lim}\limits_{h \rightarrow 0} f(a + h) = f(a)\)
64 Limits and Continuity · Level 5
To prove that sine is continuous, we need to show that \(\operatorname*{lim}\limits_{x \rightarrow a} \sin x = \sin a\) for every real number \(a\). By Exercise 63 an equivalent statement is that \(\operatorname*{lim}\limits_{h \rightarrow 0} \sin(a + h) = \sin a\) Use (6) to show that this is true.
65 Limits and Continuity · Level 5
Prove that cosine is a continuous function.
66 Limits and Continuity · Level 5
(a) Prove Theorem 4, part 3.
(b) Prove Theorem 4, part 5.

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67 Limits and Continuity · Level 5
For what values of \(x\) is \(f\) continuous? \(f(x) = \begin{cases} 0 & \quad \text{if } x \text{is rational} \\ 1 & \quad \text{if } x \text{is irrational} \end{cases}\)
68 Limits and Continuity · Level 5
For what values of \(x\) is \(g\) continuous? \(g(x) = \begin{cases} 0 & \quad \text{if } x \text{is rational} \\ x & \quad \text{if } x \text{is irrational} \end{cases}\)
69 Limits and Continuity · Level 4
Is there a number that is exactly 1 more than its cube?
70 Limits and Continuity · Level 5
If \(a\) and \(b\) are positive numbers, prove that the equation \(\dfrac{a}{x^3 + 2x^2 - 1} + \dfrac{b}{x^3 + x - 2} = 0\) has at least one solution in the interval \((-1, 1)\).
71 Limits and Continuity · Level 5
Show that the function \(f(x) = \begin{cases} x^4 \sin\left(\dfrac{1}{x}\right) & \quad \text{if } x \neq 0 \\ 0 & \quad \text{if } x = 0 \end{cases}\) is continuous on \((-\infty, \infty)\).
72 Limits and Continuity · Level 5
(a) Show that the absolute value function \(F(x) = |x|\) 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.

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73 Limits and Continuity · Level 5
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.

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