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Unformatted text preview: SECTION 2.5 CONTINUITY | | | | 119 CONTINUITY We noticed in Section 2.3 that the limit of a function as approaches can often be found simply by calculating the value of the function at . Functions with this property are called continuous at a . We will see that the mathematical definition of continuity corresponds closely with the meaning of the word continuity in everyday language. (A continuous process is one that takes place gradually, without interruption or abrupt change.) DEFINITION A function is continuous at a number a if Notice that Definition l implicitly requires three things if is continuous at a : 1. is defined (that is, a is in the domain of ) 2. exists 3. The definition says that is continuous at if approaches as x approaches a . Thus a continuous function has the property that a small change in x produces only a small change in . In fact, the change in can be kept as small as we please by keep- ing the change in sufficiently small. If is defined near (in other words, is defined on an open interval containing , except perhaps at ), we say that is discontinuous at a (or has a discontinuity at ) if is not continuous at . Physical phenomena are usually continuous. For instance, the displacement or velocity of a vehicle varies continuously with time, as does a persons height. But discontinuities do occur in such situations as electric currents. [See Example 6 in Section 2.2, where the Heaviside function is discontinuous at because does not exist.] Geometrically, you can think of a function that is continuous at every number in an interval as a function whose graph has no break in it. The graph can be drawn without removing your pen from the paper. EXAMPLE 1 Figure 2 shows the graph of a function f . At which numbers is f discontinu- ous? Why? SOLUTION It looks as if there is a discontinuity when a 1 because the graph has a break there. The official reason that f is discontinuous at 1 is that is not defined. The graph also has a break when , but the reason for the discontinuity is differ- ent. Here, is defined, but does not exist (because the left and right limits are different). So f is discontinuous at 3. What about ? Here, is defined and exists (because the left and right limits are the same). But So is discontinuous at 5. M Now lets see how to detect discontinuities when a function is defined by a formula. f lim x l 5 f x f 5 lim x l 5 f x f 5 a 5 lim x l 3 f x f 3 a 3 f 1 lim t l H t a f a f f a a f a f x f x f x f f a f x a f lim x l a f x f a lim x l a f x f f a f lim x l a f x f a f 1 a a x 2.5 N As illustrated in Figure 1, if is continuous, then the points on the graph of approach the point on the graph. So there is no gap in the curve....
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- Spring '09