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Lecture 9

Lecture 9 - 2.5.pdf

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McGill University
Mathematics & Statistics (Sci)
MATH 140
Ewa Duma

2.5 CONTINUITY We noticed in Section 2.3 that the limit of a function as x approaches a can often be found simply by calculating the value of the function at a. Functions with this property are called continuous at a. We will see that the mathematical definition of continuity corresponds MATH14closely with the meaning of the word continuity in everyday language. (A continuous process is one that takes place gradually, without interruption or abrupt change.) Continuity: 1 DEFINITION A function f is continuous at a number a if The limit of a function as x approaches a can be often found by calculating the value of the function at “a”. Functions with this property are usually called “continuouxlat a”. ▯ f▯a▯ 120 |||| CHAPTER 2 LIMITS AND DERIVATIVES DefiNAs illustrated in Figure 1, if f is continuous,a” if: V EXAMPLE 2e Where are each of the following functions discontinuous?s continuous at a: then the points ▯x, f ▯x▯▯ on the graph of f approach the point ▯a, f ▯a▯▯ on the graph1.Sf▯2▯ is defined (that is, a is in the domain of1f ) there is no gap in the curve. 2.mlim f▯x▯ exists f(a) 2 if x ▯ 0 (a) f▯x▯ ▯xla x ▯ 2 (b) f▯x▯ ▯ ▯ x y 1 if x ▯ 0 The definition requires thry=ƒthings if f is continuo2s at a:▯a▯ xlx ▯ x ▯ 2 ƒ (c)f▯x▯ ▯e definx ▯ 2 says that f is continu(d) f▯x▯ ▯ ▯x▯x▯ approaches f▯a▯ as x approaches a. ap•rf(a) is defined (that is, a is in the domain of f) f(a). Thus a c1ntinuous funcif x ▯ 2as the property that a small change in x produces only a • lim (x > a) f(x) exists small change in f▯x▯. In fact, the change in f▯x▯ can be kept as small as we please by keep- • lim (x > a) f(x) = f(a) SOLUing the change in x sufficiently small. (a) Notice thatf▯2▯is not defined, so f is discontinuous at 2. Later we’ll sef isy continuous at all other numbers. other words, f is defined on an open interval containing a, except perhaps at a), we say that f is discontinuous at a (or f has a discontinuity at a) if This implies0that a continuous funcxio(b) Heretf▯0▯ ▯ 1ris defined butll change in x produces only a small change in f(x). f is not continuous at a. As x approaches a, Physical phenomena are usulim f▯x▯ ▯ lims. For instance, the displacement or velocity of a vehicle varies continuously with time, as does a person’s height. But discontinuities If FIGURE 1ned near a (f is defined on does not exist. (See Example 8 in Section 2.2.) So f is discontinuous at 0.tion 2.2, where the we say that f is discontinuous at a (or f Heaviside function is discontinuous at 0 because limat l 0▯t▯ does not exist.] (c) Here f▯2▯ ▯ 1 is defined and Geometrically, you can think of a function that is continuous at every number in an Example: interval as a functi2n whose graph has no break in it. The graph can be drawn without removing your pen from the paper.lim ▯x ▯ 2▯▯x ▯ 1▯ ▯ lim ▯x ▯ 1▯ ▯ 3 xl2 xl2 x ▯ 2 xl2 x ▯ 2 xl2 y exists. But 1 Figure 2 shows the graph of a function f. At which numbers is f discontinu- ous? Why? lim f▯x▯ ▯ f▯2▯ xl2 SOLUTION It looks as if there is a discontinuity when a ▯ 1 because the graph has a break so fis not continuous at 2. there. The official reason that f is discontinuous at 1 is that f▯1▯ is not defined. (d) TheThe graph also has a break when a ▯ 3, but the reason for the discontinuity is differ- because limxln ▯x▯does not exist nfis an integer. (See Example 10 and Exercise 49 in 0 x Section 2.3.), f▯3▯ is defined, but limx l3f▯x▯ does not exist (because the left aMd right limits 1 2 3 4 5 are different). So f is discontinuous at 3. Figure 3 shows the graphs of the functions in Example 2. In each case the graph can’tft and Based on the graph, we can see that when x = 1 and when x l5 be dright limits are the same). Butom the paper because a hole or break or jump occurs in x =FIGURE 2 x = 1, we see that it is dthe graph. The kind of discontinuity illulim f▯x▯ ▯ f▯5▯s (a) and (c) is called removable 5, this is a jump discontinuity becausbecause we could remove the discontinuity by redefining f at just the single number 2. So f is discontinuous at 5. M [The functiont▯x▯ ▯ x ▯ 1 is continuous.] The discontinuity in part (b) is called an infi- The following graphs are more examples of jump/infinite/removable discontinuity.n part (d) are called jump discontinuities because the funNow let’s see how to detect discontinuities when a function is defined by a formula. y y y y 1 1 1 1 0 1 2 x 0 x 0 1 2 x 0 1 2 3 x
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