2.5 CONTINUITY. a x. Notice that Definition l implicitly requires three things if f is continuous at a:

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1 SECTION.5 CONTINUITY 9.5 CONTINUITY We noticed in Section.3 that the it of a function as approaches a can often be found simpl b calculating the value of the function at a. Functions with this propert are called continuous at a. We will see that the mathematical definition of continuit corresponds closel with the meaning of the word continuit in everda language. (A continuous process is one that takes place graduall, without interruption or abrupt change.) DEFINITION A function f is continuous at a number a if f f a la N As illustrated in Figure, if f is continuous, then the points, f on the graph of f approach the point a, f a on the graph. So there is no gap in the curve. ƒ approaches f(a). FIGURE f(a) =ƒ a As approaches a, FIGURE Notice that Definition l implicitl requires three things if f is continuous at a:. f a is defined (that is, a is in the domain of f ). f eists 3. la f f a la The definition sas that f is continuous at a if f approaches f a as approaches a. Thus a continuous function f has the propert that a small change in produces onl a small change in f. In fact, the change in f can be kept as small as we please b keeping the change in sufficientl small. If f is defined near a (in other words, f is defined on an open interval containing a, ecept perhaps at a), we sa that f is discontinuous at a (or f has a discontinuit at a) if f is not continuous at a. Phsical phenomena are usuall continuous. For instance, the displacement or velocit of a vehicle varies continuousl with time, as does a person s height. But discontinuities do occur in such situations as electric currents. [See Eample 6 in Section., where the Heaviside function is discontinuous at because t l H t does not eist.] Geometricall, ou can think of a function that is continuous at ever number in an interval as a function whose graph has no break in it. The graph can be drawn without removing our pen from the paper. EXAPLE Figure shows the graph of a function f. At which numbers is f discontinuous? Wh? SOLUTION It looks as if there is a discontinuit when a because the graph has a break there. The official reason that f is discontinuous at is that f is not defined. The graph also has a break when a 3, but the reason for the discontinuit is different. Here, f 3 is defined, but l3 f does not eist (because the left right its are different). So f is discontinuous at 3. What about a 5? Here, f 5 is defined l5 f eists (because the left right its are the same). But f f 5 l 5 So f is discontinuous at 5. Now let s see how to detect discontinuities when a function is defined b a formula.

2 CHAPTER LIITS AND DERIVATIVES V EXAPLE Where are each of the following functions discontinuous? (a) f (c) f if if SOLUTION (a) Notice that f is not defined, so f is discontinuous at. Later we ll see wh f is continuous at all other numbers. (b) Here f is defined but does not eist. (See Eample 8 in Section..) So f is discontinuous at. (c) Here f is defined eists. But so f is not continuous at. (d) The greatest integer function f has discontinuities at all of the integers because ln does not eist if n is an integer. (See Eample Eercise 49 in Section.3.) Figure 3 shows the graphs of the functions in Eample. In each case the graph can t be drawn without lifting the pen from the paper because a hole or break or jump occurs in the graph. The kind of discontinuit illustrated in parts (a) (c) is called removable because we could remove the discontinuit b redefining f at just the single number. [The function t is continuous.] The discontinuit in part (b) is called an infinite discontinuit. The discontinuities in part (d) are called jump discontinuities because the function jumps from one value to another. (b) (d) f l l f f l f if f if f 3 l l l l 3 (a) ƒ= -- - (b) ƒ= if if = (c) ƒ= -- if - if = (d) ƒ=[] FIGURE 3 Graphs of the functions in Eample

3 SECTION.5 CONTINUITY DEFINITION A function f is continuous from the right at a number a if f is continuous from the left at a if f f a la f f a la EXAPLE 3 At each integer n, the function f [see Figure 3(d)] is continuous from the right but discontinuous from the left because ln f n f n ln but ln f n f n ln 3 DEFINITION A function f is continuous on an interval if it is continuous at ever number in the interval. (If f is defined onl on one side of an endpoint of the interval, we underst continuous at the endpoint to mean continuous from the right or continuous from the left.) EXAPLE 4 Show that the function f s is continuous on the interval,. SOLUTION If a, then using the Limit Laws, we have f ( s ) s s s a f a (b Laws 7) (b ) (b, 7, 9) ƒ=-œ - Thus, b Definition l, f is continuous at a if a. Similar calculations show that f f l f f l - so f is continuous from the right at continuous from the left at. Therefore, according to Definition 3, f is continuous on,. The graph of f is sketched in Figure 4. It is the lower half of the circle FIGURE 4 Instead of alwas using Definitions,, 3 to verif the continuit of a function as we did in Eample 4, it is often convenient to use the net theorem, which shows how to build up complicated continuous functions from simple ones.

4 CHAPTER LIITS AND DERIVATIVES 4 THEORE If f t are continuous at a c is a constant, then the following functions are also continuous at a:. f t. f t 3. cf f 4. ft 5. if t a t PROOF Each of the five parts of this theorem follows from the corresponding Limit Law in Section.3. For instance, we give the proof of part. Since f t are continuous at a, we have f f a t t a Therefore f t f t f a t a f t a f t (b Law ) This shows that f t is continuous at a. It follows from Theorem 4 Definition 3 that if f t are continuous on an interval, then so are the functions f t, f t, cf, ft, (if t is never ) f t. The following theorem was stated in Section.3 as the Direct Substitution Propert. 5 THEORE (a) An polnomial is continuous everwhere; that is, it is continuous on,. (b) An rational function is continuous wherever it is defined; that is, it is continuous on its domain. PROOF (a) A polnomial is a function of the form P c n n c n n c c where c, c,..., c n are constants. We know that c c (b Law 7) m a m m,,..., n (b 9) This equation is precisel the statement that the function f m is a continuous function. Thus, b part 3 of Theorem 4, the function t c m is continuous. Since P is a sum of functions of this form a constant function, it follows from part of Theorem 4 that P is continuous.

5 SECTION.5 CONTINUITY 3 (b) A rational function is a function of the form f P Q where P Q are polnomials. The domain of f is D Q. We know from part (a) that P Q are continuous everwhere. Thus, b part 5 of Theorem 4, f is continuous at ever number in D. As an illustration of Theorem 5, observe that the volume of a sphere varies continuousl with its radius because the formula V r 4 3 r 3 shows that V is a polnomial function of r. Likewise, if a ball is thrown verticall into the air with a velocit of 5 ft s, then the height of the ball in feet t seconds later is given b the formula h 5t 6t. Again this is a polnomial function, so the height is a continuous function of the elapsed time. Knowledge of which functions are continuous enables us to evaluate some its ver quickl, as the following eample shows. Compare it with Eample (b) in Section.3. 3 EXAPLE 5 Find. l 5 3 SOLUTION The function f is rational, so b Theorem 5 it is continuous on its domain, which is. Therefore 3 f f l 5 3 l { 5 3} FIGURE 5 P(cos, sin ) (, ) It turns out that most of the familiar functions are continuous at ever number in their domains. For instance, Limit Law (page ) is eactl the statement that root functions are continuous. From the appearance of the graphs of the sine cosine functions (Figure 8 in Section.), we would certainl guess that the are continuous. We know from the definitions of sin cos that the coordinates of the point P in Figure 5 are cos, sin. As, we see that P approaches the point, so cos l sin l. Thus l 6 cos sin l l N Another wa to establish the its in (6) is to use the Squeeze Theorem with the inequalit sin (for ), which is proved in Section 3.3. Since cos sin, the equations in (6) assert that the cosine sine functions are continuous at. The addition formulas for cosine sine can then be used to deduce that these functions are continuous everwhere (see Eercises 56 57). It follows from part 5 of Theorem 4 that tan sin cos

6 4 CHAPTER LIITS AND DERIVATIVES _ 3π _π FIGURE 6 π _ =tan π π 3π is continuous ecept where cos. This happens when is an odd integer multiple of, so tan has infinite discontinuities when, 3, 5, so on (see Figure 6). The inverse function of an continuous one-to-one function is also continuous. (This fact is proved in Appendi F, but our geometric intuition makes it seem plausible: The graph of f is obtained b reflecting the graph of f about the line. So if the graph of f has no break in it, neither does the graph of f.) Thus the inverse trigonometric functions are continuous. In Section.5 we defined the eponential function a so as to fill in the holes in the graph of a where is rational. In other words, the ver definition of a makes it a continuous function on. Therefore its inverse function log a is continuous on,. N The inverse trigonometric functions are reviewed in Section.6. 7 THEORE their domains: The following tpes of functions are continuous at ever number in polnomials rational functions root functions trigonometric functions eponential functions inverse trigonometric functions logarithmic functions EXAPLE 6 Where is the function continuous? SOLUTION We know from Theorem 7 that the function ln is continuous for tan is continuous on. Thus, b part of Theorem 4, ln tan is continuous on,. The denominator,, is a polnomial, so it is continuous everwhere. Therefore, b part 5 of Theorem 4, f is continuous at all positive numbers ecept where. So f is continuous on the intervals,,. sin EXAPLE 7 Evaluate. l cos f ln tan SOLUTION Theorem 7 tells us that sin is continuous. The function in the denominator, cos, is the sum of two continuous functions is therefore continuous. Notice that this function is never because cos for all so cos everwhere. Thus the ratio f sin cos is continuous everwhere. Hence, b definition of a continuous function, sin l cos f f sin l cos Another wa of combining continuous functions f t to get a new continuous function is to form the composite function f t. This fact is a consequence of the following theorem.

7 SECTION.5 CONTINUITY 5 N This theorem sas that a it smbol can be moved through a function smbol if the function is continuous the it eists. In other words, the order of these two smbols can be reversed. 8 THEORE If f is continuous at b t b, then la In other words, f t f ( t ) f t f b. la Intuitivel, Theorem 8 is reasonable because if is close to a, then t is close to b, since f is continuous at b, if t is close to b, then f t is close to f b. A proof of Theorem 8 is given in Appendi F. s EXAPLE 8 Evaluate arcsin. l SOLUTION Because arcsin is a continuous function, we can appl Theorem 8: s s arcsin arcsin l l s arcsin l ( s ) ( s ) arcsin l s arcsin Let s now appl Theorem 8 in the special case where f s n, with n being a positive integer. Then f t s n t 6 f ( t ) s n t If we put these epressions into Theorem 8, we get t s sn n t so Limit Law has now been proved. (We assume that the roots eist.) 9 THEORE If t is continuous at a f is continuous at t a, then the composite function f t given b f t f t is continuous at a. This theorem is often epressed informall b saing a continuous function of a continuous function is a continuous function. PROOF Since t is continuous at a, we have t t a Since f is continuous at b t a, we can appl Theorem 8 to obtain f t f t a

8 6 CHAPTER LIITS AND DERIVATIVES which is precisel the statement that the function h f t is continuous at a; that is, f t is continuous at a. V EXAPLE 9 Where are the following functions continuous? (a) h sin (b) F ln cos SOLUTION (a) We have h f t, where t f sin _ FIGURE 7 =ln(+cos ) _6 Now t is continuous on since it is a polnomial, f is also continuous everwhere. Thus h f t is continuous on b Theorem 9. (b) We know from Theorem 7 that f ln is continuous t cos is continuous (because both cos are continuous). Therefore, b Theorem 9, F f t is continuous wherever it is defined. Now ln cos is defined when cos. So it is undefined when cos, this happens when, 3,.... Thus F has discontinuities when is an odd multiple of is continuous on the intervals between these values (see Figure 7). An important propert of continuous functions is epressed b the following theorem, whose proof is found in more advanced books on calculus. THE INTEREDIATE VALUE THEORE Suppose that f is continuous on the closed interval a, b let N be an number between f a f b, where f a f b. Then there eists a number c in a, b such that f c N. The Intermediate Value Theorem states that a continuous function takes on ever intermediate value between the function values f a f b. It is illustrated b Figure 8. Note that the value N can be taken on once [as in part (a)] or more than once [as in part (b)]. f(b) N f(b) N =ƒ f(a) =ƒ f(a) a c b a c c c b FIGURE 8 f(a) =ƒ N =N f(b) a b FIGURE 9 (a) If we think of a continuous function as a function whose graph has no hole or break, then it is eas to believe that the Intermediate Value Theorem is true. In geometric terms it sas that if an horizontal line N is given between f a f b as in Figure 9, then the graph of f can t jump over the line. It must intersect N somewhere. It is important that the function f in Theorem be continuous. The Intermediate Value Theorem is not true in general for discontinuous functions (see Eercise 44). One use of the Intermediate Value Theorem is in locating roots of equations as in the following eample. (b)

9 SECTION.5 CONTINUITY 7 V EXAPLE Show that there is a root of the equation between. SOLUTION Let f We are looking for a solution of the given equation, that is, a number c between such that f c. Therefore, we take a, b, N in Theorem. We have f f Thus f f ; that is, N is a number between f f. Now f is continuous since it is a polnomial, so the Intermediate Value Theorem sas there is a number c between such that f c. In other words, the equation has at least one root c in the interval,. In fact, we can locate a root more precisel b using the Intermediate Value Theorem again. Since f..8 f a root must lie between..3. A calculator gives, b trial error, f..78 f so a root lies in the interval.,.3. We can use a graphing calculator or computer to illustrate the use of the Intermediate Value Theorem in Eample. Figure shows the graph of f in the viewing rectangle, 3 b 3, 3 ou can see that the graph crosses the -ais between. Figure shows the result of zooming in to the viewing rectangle.,.3 b.,.. 3. _ 3..3 FIGURE _3 _. FIGURE In fact, the Intermediate Value Theorem plas a role in the ver wa these graphing devices work. A computer calculates a finite number of points on the graph turns on the piels that contain these calculated points. It assumes that the function is continuous takes on all the intermediate values between two consecutive points. The computer therefore connects the piels b turning on the intermediate piels.

10 8 CHAPTER LIITS AND DERIVATIVES.5 EXERCISES. Write an equation that epresses the fact that a function is continuous at the number 4.. If f is continuous on,, what can ou sa about its graph? 3. (a) From the graph of f, state the numbers at which f is discontinuous eplain wh. (b) For each of the numbers stated in part (a), determine whether f is continuous from the right, or from the left, or neither. f (d) The cost of a tai 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 9. If f t are continuous functions with f 3 5 l 3 f t 4, find t 3. Use the definition of continuit the properties of its to show that the function is continuous at the given number a.. f s7,. f 3 4, t 3t. h t, t 3 a a a 4 _4 _ From the graph of t, state the intervals on which t is continuous. _4 _ Use the definition of continuit the properties of its to show that the function is continuous on the given interval f, 4. t s3, 5 Eplain wh the function is discontinuous at the given number a. Sketch the graph of the function. 5. f ln 6. f,, 3 if if a a 5. Sketch the graph of a function that is continuous everwhere ecept at 3 is continuous from the left at Sketch the graph of a function that has a jump discontinuit at a removable discontinuit at 4, but is continuous elsewhere. 7. A parking lot charges $3 for the first hour (or part of an hour) $ for each succeeding hour (or part), up to a dail maimum of $. (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 their significance to someone who parks in the lot. 8. Eplain wh 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 Cit (c) The altitude above sea level as a function of the distance due west from New York Cit f e if if f f cos if if if if if f if 3 if 3 a a a a 3 8 Eplain, using Theorems 4, 5, 7, 9, wh the function is continuous at ever number in its domain. State the domain.. F. G s

11 SECTION.5 CONTINUITY 9 3. R s L t e 5t cos t G t ln t 4 8. h sin F sin H cos(e s ) 4. For what value of the constant c is the function f continuous on,? f c 3 c if if ; 9 3 Locate the discontinuities of the function illustrate b graphing Use continuit to evaluate the it Show that f is continuous on, 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 l e e 5 s l4 s5 f if s if f sin if 4 cos if 4 f f s 3 f e 4. The gravitational force eerted b the earth on a unit mass at a distance r from the center of the planet is F r Gr R 3 G r 3. if if if if if 3 if 3 if if if ln tan sin sin l arctan 4 l 3 6 if r R if r R where is the mass of the earth, R is its radius, G is the gravitational constant. Is F a continuous function of r? 4. Find the values of a b that make f continuous everwhere. 43. Which of the following functions f has a removable discontinuit at a? If the discontinuit is removable, find a function t that agrees with f for a is continuous at a. (a) f 4, (b) f 3, (c) f sin, 44. Suppose that a function f is continuous on [, ] ecept at.5 that f f 3. Let N. Sketch two possible graphs of f, one showing that f might not satisf the conclusion of the Intermediate Value Theorem one showing that f might still satisf the conclusion of the Intermediate Value Theorem (even though it doesn t satisf the hpothesis). 45. If f sin, show that there is a number c such that f c. 46. Suppose f is continuous on, 5 the onl solutions of the equation f 6 are 4. If f 8, eplain wh f Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval ,, 48. s 3, 49. cos,, 5. ln e, 5 5 (a) Prove that the equation has at least one real root. (b) Use our calculator to find an interval of length. that contains a root. 5. cos 3 5. ln 3 ; (a) Prove that the equation has at least one real root. (b) Use our graphing device to find the root correct to three decimal places. 53. f e. 4 a b 3 a b a a a if if 3 if 3, 54. arctan,

12 3 CHAPTER LIITS AND DERIVATIVES 55. Prove that f is continuous at a if onl if 56. To prove that sine is continuous, we need to show that sin sin a for ever real number a. B Eercise 55 an equivalent statement is that Use (6) to show that this is true. 57. Prove that cosine is a continuous function. 58. (a) Prove Theorem 4, part 3. (b) Prove Theorem 4, part For what values of is f continuous? 6. For what values of is t continuous? 6. f a h f a h l sin a h sin a h l f t if is rational if is irrational if is rational if is irrational Is there a number that is eactl more than its cube? 6. If a b are positive numbers, prove that the equation has at least one solution in the interval,. 63. Show that the function a 3 b 3 f 4 sin is continuous on,. if if F 64. (a) Show that the absolute value function is continuous everwhere. (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 countereample. 65. A Tibetan monk leaves the monaster at 7: A takes his usual path to the top of the mountain, arriving at 7: P. The following morning, he starts at 7: A at the top takes the same path back, arriving at the monaster at 7: P. Use the Intermediate Value Theorem to show that there is a point on the path that the monk will cross at eactl the same time of da on both das f FIGURE LIITS AT INFINITY; HORIZONTAL ASYPTOTES In Sections..4 we investigated infinite its vertical asmptotes. There we let approach a number the result was that the values of became arbitraril large (positive or negative). In this section we let become arbitraril large (positive or negative) see what happens to. Let s begin b investigating the behavior of the function f defined b f as becomes large. The table at the left gives values of this function correct to si decimal places, the graph of f has been drawn b a computer in Figure. As grows larger larger ou can see that the values of f get closer closer to. In fact, it seems that we can make the values of f as close as we like to b taking sufficientl large. This situation is epressed smbolicall b writing = l = - +