Piecewise Continuous Functions - Dartmouth College

1 Piecewise Continuous FunctionsLeft and Right LimitsIn our last lecture, we discussed the trigonometric Functions tangent, cotangent, secant, and cosecant. All ofthese Functions differed from sine and cosine in that they were not defined at all real numbers. At the pointsat which these Functions were not defined, we found vertical asymptotes. As you may recall, a functionf(x)has a positive left vertical asymptote, for instance, at a pointaif, asxapproachesafrom the negative, orleft, side, the function either becomes more and more positive. Another way to say this is that for everypositive numberb, there is some distancedsuch that ifxis within a distance ofdofaon its left side, wecan guarantee thatf(x) will be bigger thanb.

2 For example, we know that the functionf(x) =x 2has apositive left vertical asymptote atx= 0. If say, we wanted to guarantee thatf(x) is greater than 100, wecould do this by choosing a value ofxwhich is within of 0, that is, < x <0. If we want to guaranteethatf(x) is greater than 10000, we would choose a value ofxbetween and 0. This idea of guaranteedlargeness is at the heart of vertical we want to extend this idea a bit further, and introduce the idea of left and right limits. Letf(x)be some real valued function. We say thatf(x) has a left limit ofSatx=aif asxapproachesafrom thenegative side,f(x) approachesS. We write this in limit notation in the following way:limx a f(x) = , we say thatf(x) has a right limit ofRatx=aif asxapproachesafrom the positive side,f(x)approachesR.

3 The limit notation for a right limit islimx a+f(x) = left and right vertical asymptotes, we have the notion of guaranteed largeness (or smallness, if wehave a negative vertical asymptote). For left and right limits, we have a notion of guaranteed closeness. Iff(x) has a right limit ofRatx=a, what we are saying is that, if we want to guarantee thatf(x) is goingto be within some distancebofR(in other words,R b < f(x)< R+b), no matter how smallbhappens tobe, we can find some numberdsuch that ifxis withindofaon its positive side, thenf(x) will be withinbofR. So, for instance, if is well-known among mathematicians thatf(x) =x2has a right limit of 0 atx= we want to guarantee thatf(x) is going to be within of 0, that is, < f(x)< Tomake this guarantee, we choosexto be within of 0 on its right side, that is, 0< x < If we want toguarantee thatf(x) is within of 0, then we choosexto be within of 0.

4 No matter how small abound we put onf(x), we can always guarantee thatf(x) will be within that bound by placing a suitablysmall bound onx. This is the essence of left and right and Piecewise Continuous FunctionsIn the example above, we noted thatf(x) =x2has a right limit of 0 atx= 0. It also has a left limit of 0atx= 0. This should make intuitive sense to you if you draw out the graph off(x) =x2: as we approachx= 0 from the negative side,f(x) gets closer and closer to 0. It is not surprising thatf(x) =x2has botha left limit and a right limit of 0 atx= 0, sincef(0) = 0, and the graph of this function has no breaks init, that is, if we draw the graph ofx2with a pen, as we pass throughx= 0, we never have to lift the is the intuitive concept of continuity.

5 Mathematically, we define continuity using (x) be a real valued function. Suppose thatf(x) has both a left limitSand a right limitRatx=a. Suppose further thatf(x) is defined atx=a, and thatRandSare both equal tof(a). We writethis in limit notation aslimx a f(x) =f(a) = limx a+f(x).We then say thatf(x) is Continuous atx=a. In other words, if we approachx=afrom either the left orthe right,f(x) approachesf(a). The functionf(x) =x2is Continuous atx= 0 by this definition. It is alsocontinuous at every other point on the real line by this definition. If a function is Continuous at every pointin its domain, we call it a Continuous function. The following Functions are all Continuous :1 polynomial Functions sine and cosine exponential and generalized exponential Functions any stretches or shifts of the Functions aboveTechnically, negative power Functions and the other trigonometric Functions are also Continuous everywhereon their domains, and yet these Functions have breaks in them: you cannot draw the graphs of these functionswithout lifting your pen at least once.

6 This is because these Functions are not defined everywhere, and thebreaks in the graphs only occur where the Functions are undefined. In this sense, our mathematical definitionof continuity does not match our intuitive concept of continuity. This is a very subtle point, and you shouldnot expect to be tested on it, nor will you be marked off if you say that negative power Functions or theother four trigonometric Functions are far in this class, we have not studied truly discontinuous Functions , but that changes today. First,consider the following function:f(x) ={2x+ 3x6= 14x= we draw the graph of this function, we see a line with a hole in it atx= 1, and, above the hole, a pointat ( 1,4).}

7 Consider the left and right limits off(x) atx= 1. Clearly, asxapproaches 1 from the left,f(x) approaches 2 ( 1) + 3 = 1. In other words, as we travel along the graph off(x) from the left tox= 1, the graph gets closer and closer to the point ( 1,1), which is where the hole in the line is. Thesame thing occurs if we move along the graph from the right tox= 1. Thus, in limit notation, we havethatlimx 1 f(x) = 1 andlimx 1+f(x) = the left limit and the right limit off(x) atx= 1 are equal, but not equal tof( 1), which is defined tobe 4. Thereforef(x) is discontinuous atx= 1. This example fits our intuitive understanding of continuityas well: in order to draw the graph off(x), we need to lift our pen atx= 1 so that we can draw the point( 1,4).

8 The functionf(x) atx= 1 is an example of a function which has a both a left limit and a right limitat a point, and while the left limit and the right limit do not equal the function at that point, they do equaleach other. In general, iff(x) has both a left limit and a right limit atx=a, and the left limit and theright limit both equalL, then we simply say thatf(x) has a limitLatx=a, and we writelimx af(x) = , in our example above, we would writelimx 1f(x) = course, there exist Functions for which the left limit and the right limit do exist, but they do not equaleach other. For example, takeg(x) ={ x+ 1x <0x 1x the behavior ofg(x) atx= 0.}

9 Asxapproaches 0 from the left, the function behaves like the linearfunction x+ 1, sog(x) approaches 1. Asxapproaches 0 from the right, the function behaves like anotherlinear function,x 1, and sog(x) approaches 1. Thus we have thatlimx 0 g(x) = 16= 1 = limx 0+g(x).This function is clearly discontinuous atx= 0, since the left and right limits do not equal each other, letalone both of them equalling the value of the function at 0. The right limit does equal the value of the2function atx= 0 , however, and so we say thatg(x) is right Continuous atx= 0. In general, iff(x) has aright limit atx=aand that limit equalsf(a), then we say thatf(x) is right Continuous atx=a, and iff(x) has a left limit atx=aand that limit equalsf(a), then we say thatf(x) is left Continuous atx= you see why iff(x) is both left Continuous and right Continuous atx=athenf(x) is Continuous atx=a?

10 The Functions that we have been using as examples above, which are Continuous everywhere except ata small number of points, are called Piecewise Continuous Functions . We usually write Piecewise continuousfunctions by defining them case by case on different intervals. For example,h(x) = x2+ 4x+ 3x < 3x+ 3 3 x <1 2x= 1ex1< x ln 2e xx >ln 2is a Piecewise Continuous function. As an exercise, sketch out this function and decide where it is Continuous ,left Continuous , and right Continuous . Pay special attention to the behavior ofh(x) atx= is one final point: iff(x) is not Continuous atx=a, thenf(x) cannot have a derivative atx=aeither. Intuitively, this makes sense , because it makes no sense to define a tangent line to a function at apoint where it is discontinuous.