Continuity and One-Sided Limits Notes

CONTINUIТY AND ONE-SIDED LIMITS NOТES ASSIGNMENT Objectives: ■ Determine continuity at а point and continuity оп an open interval. ■ Determine one-sided limits and continuity оп а closed interval. Continuity at а Point and Continuity оп an Open lnterval ln mathematics, the term continuous has much the same meaning as it has in everyday usage. lnformally, to say that а function f is continuous at х = с means that there is по interruption in the graph of / at с. That is, its graph is unbroken at с and there are по holes, jumps, or gaps. The graphs shown below identify three values of х at which the graph of / is not continuous. At all other points in the interval (а, Ь), the graph of / is uninterrupted and continuous. у у ( ) _,. (i) (з) lf( limf(.x) , lotJr, /.J и ои s А--+С does not exist. li111/(x) -1:f(c) Х➔С ��: 1 1 1 1 ------------- х а с ь 1 1 ----------х а с ь --+-1'----➔------+--Х а с ь Tl1ree coпditions exist fог whic\1 the grap\1 of / is поt continuous at х = с. lt appears that continuity at х = с сап Ье destroyed Ьу any one of the following conditions. 1. The function is not defined at х = с. 2. The limit of f(x) does not exist at х = с. 3. The limit of f(x) exists at х = с, but it is not equal to /(с). �(попе of the three conditions above is true, the function f is called continuous at с, as indicated in the following important definition. DEFINIТION OF CONТINUIТY Continuity at а Point: conditions are met. А ftшctionf is continuous at с if the following three 1. /(с) is defined. 2. lim f(x) exists. х➔с 3. lim/(x) = /(с) х➔с Continuity оп ап Ореп lnterval: А fuпction is continuous оп an open interval (а, Ь) if it is continuous at each point in the interval. А function that is continuous on the епtiге 1·eal line ( - оо, оо) is every,vhere coпtinuous. Consider an open interval / that contains а real number с. lf а function f is defined on / (except possiЫy at с), and f is not continuous at с, then f is said to have а discontinuity at с. Discontinuities fall into two categories: removaЫe and nonremovaЫe. А discontinuity at с is called removaЬle if f can Ье made continuous Ьу appropriately defining (or redefiningf(c)). For instance, the functioris shown in Figures 1.26{а) and (с) have removaЫe discontinuities at с and the function shown in Figure 1.2б(Ь) has а nonremovaЫe discontinuity at с. у -+-+----t---+-t а с ь la) RcmovaЫc d�coлlinuit / а thl Figure 1.26 Examples: Graph each function and discuss the continuity of each function. 1 а. f(x) = -х х 2 -1 Ь. g(x) = : х-1 �lf( �(У-..)-:.. Х +1 tJot.)' (2&.,А()J.Д е.с....,; D •�a>tJ,, ..,ц 1-ry А7 X.�D d. у= sinx [ 0 � 1 �u t>tA .S (-r,-, с.о) Q.Е..ц ЫАf, � D•��, IJLl(T'/ д.,, J(&:' sinx . е. g(x) = х 2+з 1, .. ,.J х + 1, х :5 О с. h(х ) _ - { е х, х > о One-Sided Limits and Continuity оп а Closed lnterval То understand continuity оп а closed interval, you first need to look at а different type of limit called а one-sided limit. For example, the limit from the right (or right-hand limit) means that х approaches с from values greater than с This limit is denoted as: у lim f(x) = L х appmache (' rгuш tl1e гighl. х---+с + ------------х с<х Limit f"гom гight Similarly, the limit from the left (or left-hand limit) means that х approaches с from values less than с This limit is denoted as: у lim f(x) = L х approacheь с Гrom the let't. Х➔С- ------+------х с>х Limit fюm \eft Examples: Graph each function and find the limit of each а. 1- о lim ✓4 - х 2 х-+-2 + J ь. с. lim [х] lim [х] r:---:::-lo � х-+о- х-+о + -1. :. �-r IJ.Jr�fl.,. п St-;c..1-\ ,�� у'\ L Х • .-о D ,.:,uffi �5 WGfly � (�DJJfr 11\"i l J.J11:::, � �) When the limit from the left is not equal to the limit from the right, the (two-sided) limit does not exist. pqc, s� P�&.s 3, 4, s✓ {р THEOREM 1.10 ТНЕ EXISTENCE OF А LIMIТ Let/be а function and let с and L Ье real numbers. The liшit of/(x) as х approaches с is L if and only if lim_ .f(x) х--н· =L and lim I(x) х�с + = L. DEFINIТION OF CONТINUITY ON А CLOSED INTERVAL А function J is continuous on the closed interval [а, Ь] if it is continuous on the open interval (а, Ь) and lim J(x) х➔а + = J(a) and lim J(x) ь- х➔ = J(b). The function.f is continuous from the right at а and continuous from the left at Ь (see Figure 1.31). h 11 С 1i11 Г� nct о J � 1nttn1I Figure 1.31 Examples: Discuss the continuity of each • а. f(x) = Гх Ь. g(x) = ✓2 - х с. f(x) = ✓1 - х 2 1)€.fl� f=\i '] l.OA-AttvUOU$ L..o l\.lГtt-J tA D t,,U ou IJJ� L (- , z] �) LDм. ь JJ ' tv�eil)J/, [--1) -,