ContinuityPolynomials, for example P(x) = cnxn+ · · · + c1x + c0,have the following nice property:limx→aP(x) = P(a).In other words what should happen is what does happen.More generally any function which has this property issaid to be continuous.Three things need to happen for f(x) to be continuous atx = a:1. limx→af(x) exists.2. f(a) is defined.3. The two preceding values match.A function is discontinuous if it is not continuous. There areseveral different ways for a function to be discontinuous ata point.Removable discontinuityIf a function is not continuous at x = a and limx→af(x)exists, then it is a removable discontinuity in that by(re)defining the function at x = a we can make thefunction continuous.Jump discontinuityIf the left and right limits at x = a exist, but do not agree.(When drawing the pen has to jump at that point to keepgoing.) Cannot patch by redefining f(x) at x = a.Often times jump discontinuities occur for piecewisefunctions where we are gluing together. If we want toavoid a jump discontinuity where we are gluing, then weneed to ensure that the two sides line up correctly.Other discontinuitiesOther discontinuities happen when the limit does not existfrom one or both sides, including (possibly) asymptotes orfunctions similar to sin(1x).A function is continuous if it is continuous at every point inits domain. Examples include• Polynomials, and rational functions (wheredenominator 6= 0)• ex, sin x, cos x, arctan x• ln |x|, tan x, sec x (away from asymptotes)We can combine continuous functions to make newcontinuous functions. If f(x) and g(x) are continuous thenso also are:• f(x) + g(x)• κf(x)• f(x)g(x)•f(x)g(x)(where g(x) 6= 0)• f(g(x)) (composition)If we know a function is continuous, then we have theadvantage of being able to take limits easily sincelimx→af(x) = f(a). So if you see you are working with acontinuous function and taking a limit then we shouldplug in the limiting point; if we get a value, that is theanswer.Intermediate Value TheoremIf f(x) is continuous for a 6 x 6 b, then for any t betweenf(a) and f(b) there is some c between a and b withf(c) = t.baf(a)f(b)Warning: The intermediate value theorem does not saythat the output of the function stays between f(a) and f(b).For example in the figure above we see that we dip belowthe value f(b) in several places.One way we can use the intermediate value theorem islooking for roots (in other words where f(x) = 0. Forexample if a function is continuous, f(a) < 0 and f(b) > 0then it must be the case that somewhere between a and bmust be at least one root. On the other hand if bothf(a) > 0 and f(b) > 0 then we cannot make anyconclusion, it is possible that the function could have rootsby going down and then back up.Problems1. Find a and b so that f(x) is a continuous function.f(x) =3x2if x < −1ax + b if − 1 6 x 6 14x2+ 1 if x > 12. Suppose that f(x) is as shown in the figure below.1 2 3 4−2−112(a) For which values of a with 0 < a < 4 doeslimx→af(x) not exist?(b) How should the function be modified to delete aremovable discontinuity?(c) Is the function (x − 1)f(x) continuous at x = 1?Justify your answer.3. Find limx→2sin(ex2+ 23) + x5arctan(x) − 17 cos(sin(x))x4+ 5 + sin2(ex+ e−x) + ln(x2+ 1)4. Let P(x) = 18x3− 63x2+ 67x − 20.(a) Does P(x) have a root in the interval 0 6 x 6 1?(b) Does P(x) have a root in the interval 1 6 x 6 2?5. Show that there is some value x with 0 6 x 6 1 andcos x = x.6. Determine the value(s) of a so that the followingpiece-wise function will be continuous.f(x) =e2x+ 10 sin2(x) if x < a,7ex− 10 cos2(x) if x > a7. Find values of a, b, c which are positive and so that thefollowing function is continuous.f(x) =1 − cos(ax)x2if x < 0;8 if 0 6 x 6 1;(x − b)(x + c)x2+ 3x − 4if 1 <