2.5E: Exercises for Section 2.4

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Jun 24, 2021
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- 2.5: Continuity
- 2.6: The Precise Definition of a Limit

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For exercises 1 - 8, determine the point(s), if any, at which each function is discontinuous. Classify any discontinuity as jump, removable, infinite, or other.

1) $𝑓 (𝑥) = 1 \sqrt 𝑥$

Answer: The function is defined for all $𝑥$ in the interval $(0, \infty)$ .

2) $𝑓 (𝑥) = 2 𝑥 2 + 1$

3) $𝑓 (𝑥) = 𝑥 𝑥 2 - 𝑥$

Answer: Removable discontinuity at $𝑥 = 0$ ; infinite discontinuity at $𝑥 = 1$ .

4) $𝑔 (𝑡) = 𝑡 - 1 + 1$

5) $𝑓 (𝑥) = 5 𝑒 𝑥 - 2$

Answer: Infinite discontinuity at $𝑥 = l n 2$

6) $𝑓 (𝑥) = | 𝑥 - 2 | 𝑥 - 2$

7) $𝐻 (𝑥) = t a n 2 𝑥$

Answer: Infinite discontinuities at $𝑥 = (2 𝑘 + 1) 𝜋 4$ , for $𝑘 = 0, \pm 1, \pm 2, \pm 3, \dots$

8) $𝑓 (𝑡) = 𝑡 + 3 𝑡 2 + 5 𝑡 + 6$

For exercises 9 - 14, decide if the function continuous at the given point. If it is discontinuous, what type of discontinuity is it?

9) $2 𝑥 2 - 5 𝑥 + 3 𝑥 - 1$ at $𝑥 = 1$

Answer: No. It is a removable discontinuity.

10) $ℎ (𝜃) = s i n 𝜃 - c o s 𝜃 t a n 𝜃$ at $𝜃 = 𝜋$

11) $𝑔 (𝑢) = ⎧ { {⎨ { {⎩ 6 𝑢 2 + 𝑢 - 2 2 𝑢 - 1, i f 𝑢 \neq 12 72, i f 𝑢 = 12$ , at $𝑢 = 12$

Answer: Yes. It is continuous.

12) $𝑓 (𝑦) = s i n (𝜋 𝑦) t a n (𝜋 𝑦)$ , at $𝑦 = 1$

13) $𝑓 (𝑥) = {𝑥 2 - 𝑒 𝑥, i f 𝑥 < 0 𝑥 - 1, i f 𝑥 \geq 0$ , at $𝑥 = 0$

Answer: Yes. It is continuous.

14) $𝑓 (𝑥) = {𝑥 s i n (𝑥), i f 𝑥 \leq 𝜋 𝑥 t a n (𝑥), i f 𝑥 > 𝜋$ , at $𝑥 = 𝜋$

In exercises 15 - 19, find the value(s) of $𝑘$ that makes each function continuous over the given interval.

15) $𝑓 (𝑥) = {3 𝑥 + 2, i f 𝑥 < 𝑘 2 𝑥 - 3, i f 𝑘 \leq 𝑥 \leq 8$

Answer: $𝑘 = - 5$

16) $𝑓 (𝜃) = {s i n 𝜃, i f 0 \leq 𝜃 < 𝜋 2 c o s (𝜃 + 𝑘), i f 𝜋 2 \leq 𝜃 \leq 𝜋$

17) $𝑓 (𝑥) = ⎧ { {⎨ { {⎩ 𝑥 2 + 3 𝑥 + 2 𝑥 + 2, i f 𝑥 \neq - 2 𝑘, i f 𝑥 = - 2$

Answer: $𝑘 = - 1$

18) $𝑓 (𝑥) = {𝑒 𝑘 𝑥, i f 0 \leq 𝑥 < 4 𝑥 + 3, i f 4 \leq 𝑥 \leq 8$

19) $𝑓 (𝑥) = {\sqrt 𝑘 𝑥, i f 0 \leq 𝑥 \leq 3 𝑥 + 1, i f 3 < 𝑥 \leq 10$

Answer: $𝑘 = 163$

In exercises 20 - 21, use the Intermediate Value Theorem (IVT).

20) Let $ℎ (𝑥) = {3 𝑥 2 - 4, i f 𝑥 \leq 2 5 + 4 𝑥, i f 𝑥 > 2$ Over the interval $[0, 4]$ , there is no value of $𝑥$ such that $ℎ (𝑥) = 10$ , although $ℎ (0) < 10$ and $ℎ (4) > 10$ . Explain why this does not contradict the IVT.

21) A particle moving along a line for time $𝑡$ has a position function $𝑠 (𝑡)$ , which is continuous. Assume $𝑠 (2) = 5$ and $𝑠 (5) = 2$ . Another particle moves such that its position is given by $ℎ (𝑡) = 𝑠 (𝑡) - 𝑡$ . Explain why there must be a value $𝑐$ for $2 < 𝑐 < 5$ such that $ℎ (𝑐) = 0$ .

Answer: Since both $𝑠$ and $𝑦 = 𝑡$ are continuous everywhere, then $ℎ (𝑡) = 𝑠 (𝑡) - 𝑡$ is continuous everywhere and, in particular, it is continuous over the closed interval [ $2, 5$ ]. Also, $ℎ (2) = 3 > 0$ and $ℎ (5) = - 3 < 0$ . Therefore, by the IVT, there is a value $𝑥 = 𝑐$ such that $ℎ (𝑐) = 0$ .

22) [T] Use the statement “The cosine of $𝑡$ is equal to $𝑡$ cubed."

a. Write a mathematical equation of the statement.

b. Prove that the equation in part a. has at least one real solution.

c. Use a calculator to find an interval of length $0.01$ that contains a solution.

23) Apply the IVT to determine whether $2 𝑥 = 𝑥 3$ has a solution in one of the intervals [ $1.25, 1.375$ ] or [ $1.375, 1.5$ ]. Briefly explain your response for each interval.

Answer: The function $𝑓 (𝑥) = 2 𝑥 - 𝑥 3$ is continuous over the interval [ $1.25, 1.375$ ] and has opposite signs at the endpoints.

24) Consider the graph of the function $𝑦 = 𝑓 (𝑥)$ shown in the following graph.

$A diagram illustrating the intermediate value theorem. There is a generic continuous curved function shown over the interval [a,b]. The points fa. and fb. are marked, and dotted lines are drawn from a, b, fa., and fb. to the points (a, fa.) and (b, fb.). A third point, c, is plotted between a and b. Since the function is continuous, there is a value for fc. along the curve, and a line is drawn from c to (c, fc.) and from (c, fc.) to fc., which is labeled as z on the y axis.$

a. Find all values for which the function is discontinuous.

b. For each value in part a., state why the formal definition of continuity does not apply.

c. Classify each discontinuity as either jump, removable, or infinite.

25) Let $𝑓 (𝑥) = {3 𝑥, i f 𝑥 > 1 𝑥 3, i f 𝑥 < 1$ .

a. Sketch the graph of $𝑓$ .

b. Is it possible to find a value $𝑘$ such that $𝑓 (1) = 𝑘$ , which makes $𝑓 (𝑥)$ continuous for all real numbers? Briefly explain.

Answer

$A graph of the given piecewise function containing two segments. The first, x^3, exists for x < 1 and ends with an open circle at (1,1). The second, 3x, exists for x > 1. It beings with an open circle at (1,3).$

b. It is not possible to redefine $𝑓 (1)$ since the discontinuity is a jump discontinuity.

26) Let $𝑓 (𝑥) = 𝑥 4 - 1 𝑥 2 - 1$ for $𝑥 \neq - 1, 1$ .

a. Sketch the graph of $𝑓$ .

b. Is it possible to find values $𝑘 1$ and $𝑘 2$ such that $𝑓 (- 1) = 𝑘$ and $𝑓 (1) = 𝑘 2$ , and that makes $𝑓 (𝑥)$ continuous for all real numbers? Briefly explain.

27) Sketch the graph of the function $𝑦 = 𝑓 (𝑥)$ with properties i. through vii.

i. The domain of $𝑓$ is ( $- \infty, + \infty$ ).

ii. $𝑓$ has an infinite discontinuity at $𝑥 = - 6$ .

iii. $𝑓 (- 6) = 3$

iv. $l i m 𝑥 \to - 3 - 𝑓 (𝑥) = l i m 𝑥 \to - 3 + 𝑓 (𝑥) = 2$

v. $𝑓 (- 3) = 3$

vi. $𝑓$ is left continuous but not right continuous at $𝑥 = 3$ .

vii. $l i m 𝑥 \to - \infty 𝑓 (𝑥) = - \infty$ and $l i m 𝑥 \to + \infty 𝑓 (𝑥) = + \infty$

Answer

Answers may vary; see the following example:

$A graph of a piecewise function with several segments. The first is an increasing line that exists for x < -8. It ends at an open circle at (-8,-8). The second is an increasing curve that exists from -8 <= x < -6. It begins with a closed circle at (-8, 0 ) and goes to infinity as x goes to -6 from the left. The third is a closed circle at the point (-6, 3). The fourth is a line that exists from -6 < x <= 3. It begins with an open circle at (-6, 2) and ends with a closed circle at (3,2). The fifth is an increasing line starting with an open circle at (3,3). It exists for x > 3.$

28) Sketch the graph of the function $𝑦 = 𝑓 (𝑥)$ with properties i. through iv.

i. The domain of $𝑓$ is [ $0, 5$ ].

ii. $l i m 𝑥 \to 1 + 𝑓 (𝑥)$ and $l i m 𝑥 \to 1 - 𝑓 (𝑥)$ exist and are equal.

iii. $𝑓 (𝑥)$ is left continuous but not continuous at $𝑥 = 2$ , and right continuous but not continuous at $𝑥 = 3$ .

iv. $𝑓 (𝑥)$ has a removable discontinuity at $𝑥 = 1$ , a jump discontinuity at $𝑥 = 2$ , and the following limits hold: $l i m 𝑥 \to 3 - 𝑓 (𝑥) = - \infty$ and $l i m 𝑥 \to 3 + 𝑓 (𝑥) = 2$ .

In exercises 29 - 30, suppose $𝑦 = 𝑓 (𝑥)$ is defined for all $𝑥$ . For each description, sketch a graph with the indicated property.

29) Discontinuous at $𝑥 = 1$ with $l i m 𝑥 \to - 1 𝑓 (𝑥) = - 1$ and $l i m 𝑥 \to 2 𝑓 (𝑥) = 4$

Answer

Answers may vary; see the following example:

$The graph of a piecewise function with two parts. The first part is an increasing curve that exists for x < 1. It ends at (1,1). The second part is an increasing line that exists for x > 1. It begins at (1,3).$

30) Discontinuous at $𝑥 = 2$ but continuous elsewhere with $l i m 𝑥 \to 0 𝑓 (𝑥) = 12$

Determine whether each of the given statements is true. Justify your response with an explanation or counterexample.

31) $𝑓 (𝑡) = 2 𝑒 𝑡 - 𝑒 - 𝑡$ is continuous everywhere.

Answer: False. It is continuous over ( $- \infty, 0$ ) ∪ ( $0, \infty$ ).

32) If the left- and right-hand limits of $𝑓 (𝑥)$ as $𝑥 \to 𝑎$ exist and are equal, then $𝑓$ cannot be discontinuous at $𝑥 = 𝑎$ .

33) If a function is not continuous at a point, then it is not defined at that point.

Answer: False. Consider $𝑓 (𝑥) = {𝑥, i f 𝑥 \neq 0 4, i f 𝑥 = 0$ .

34) According to the IVT, $c o s 𝑥 - s i n 𝑥 - 𝑥 = 2$ has a solution over the interval [ $- 1, 1$ ].

35) If $𝑓 (𝑥)$ is continuous such that $𝑓 (𝑎)$ and $𝑓 (𝑏)$ have opposite signs, then $𝑓 (𝑥) = 0$ has exactly one solution in [ $𝑎, 𝑏$ ].

Answer: False. Consider $𝑓 (𝑥) = c o s (𝑥)$ on [ $- 𝜋, 2 𝜋$ ].

36) The function $𝑓 (𝑥) = 𝑥 2 - 4 𝑥 + 3 𝑥 2 - 1$ is continuous over the interval [ $0, 3$ ].

37) If $𝑓 (𝑥)$ is continuous everywhere and $𝑓 (𝑎), 𝑓 (𝑏) > 0$ , then there is no root of $𝑓 (𝑥)$ in the interval [ $𝑎, 𝑏$ ].

Answer: False. The IVT does not work in reverse! Consider $(𝑥 - 1) 2$ over the interval [ $- 2, 2$ ].

[T] The following problems consider the scalar form of Coulomb’s law, which describes the electrostatic force between two point charges, such as electrons. It is given by the equation $𝐹 (𝑟) = 𝑘 𝑒 | 𝑞 1 𝑞 2 | 𝑟 2$ , where $𝑘 𝑒$ is Coulomb’s constant, $𝑞 𝑖$ are the magnitudes of the charges of the two particles, and $𝑟$ is the distance between the two particles.

38) To simplify the calculation of a model with many interacting particles, after some threshold value $𝑟 = 𝑅$ , we approximate $𝐹$ as zero.

a. Explain the physical reasoning behind this assumption.

b. What is the force equation?

c. Evaluate the force $𝐹$ using both Coulomb’s law and our approximation, assuming two protons with a charge magnitude of $1.6022 \times 10 - 19$ coulombs (C), and the Coulomb constant $𝑘 𝑒 = 8.988 \times 109 𝑁 𝑚 2 / 𝐶 2$ are 1 m apart. Also, assume $𝑅 < 1$ m. How much inaccuracy does our approximation generate? Is our approximation reasonable?

d. Is there any finite value of R for which this system remains continuous at R?

39) Instead of making the force $0$ at $𝑅$ , we let the force be $10 - 20$ for $𝑟 \geq 𝑅$ . Assume two protons, which have a magnitude of charge $1.6022 \times 10 - 19 𝐶$ , and the Coulomb constant $𝑘 𝑒 = 8.988 \times 109 𝑁 𝑚 2 / 𝐶 2$ . Is there a value $𝑅$ that can make this system continuous? If so, find it.

Answer: $𝑅 = 0.0001519$ m

Recall the discussion on spacecraft from the chapter opener. The following problems consider a rocket launch from Earth’s surface. The force of gravity on the rocket is given by $𝐹 (𝑑) = - 𝑚 𝑘 / 𝑑 2$ , where m is the mass of the rocket, $𝑑$ is the distance of the rocket from the center of Earth, and $𝑘$ is a constant.

40) [T] Determine the value and units of $𝑘$ given that the mass of the rocket on Earth is 3 million kg. (Hint: The distance from the center of Earth to its surface is 6378 km.)

41) [T] After a certain distance $𝐷$ has passed, the gravitational effect of Earth becomes quite negligible, so we can approximate the force function by $𝐹 (𝑑) = {- 𝑚 𝑘 𝑑 2, i f 𝑑 < 𝐷 10, 000, i f 𝑑 \geq 𝐷$ . Find the necessary condition $𝐷$ such that the force function remains continuous.

Answer: $𝐷 = 63.78$ km

42) As the rocket travels away from Earth’s surface, there is a distance D where the rocket sheds some of its mass, since it no longer needs the excess fuel storage. We can write this function as $𝐹 (𝑑) = ⎧ { { {⎨ { { {⎩ - 𝑚 1 𝑘 𝑑 2, i f 𝑑 < 𝐷 - 𝑚 2 𝑘 𝑑 2, i f 𝑑 \geq 𝐷$ . Is there a value of $𝐷$ such that this function is continuous, assuming $𝑚 1 \neq 𝑚 2$ ?

In Exercises 43 - 44, prove each function is continuous everywhere.

43) $𝑓 (𝜃) = s i n 𝜃$

Answer: For all values of $𝑎$ , $𝑓 (𝑎)$ is defined, $l i m 𝜃 \to 𝑎 𝑓 (𝜃)$ exists, and $l i m 𝜃 \to 𝑎 𝑓 (𝜃) = 𝑓 (𝑎)$ . Therefore, $𝑓 (𝜃)$ is continuous everywhere.

44) $𝑔 (𝑥) = | 𝑥 |$

45) Where is $𝑓 (𝑥) = {0, i f 𝑥 i s i r r a t i o n a l 1, i f 𝑥 i s r a t i o n a l$ continuous?

Answer: Nowhere

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