University:
Washington State University
Course:
MATH 106 | College Algebra
Academic year:

2023
Views:

363

Pages:

5
Author:

NeatGold

CONTINUITYA function is continuous at a pointx = a if its graph isn't brokeninthe neighborhood of the point, i.e,n Lim a f(x) = x-a Lim f(x) = f(a)Afunctioniscontinuousinaninterval(a, b) if :2-C lim f(x) = f(c) for all C E (a, b)In[a,b]itiscontinuousifef(x) is cont in (a, b)f(x) is right cont x = a - x-0+ Lim f(x) = f(a)f(x) is left cont x = b - x-5 him f(x) = f(b).Types of Discontinuity-Removable (Limit exists)Isolated PointMissing Pointaa-Non Removable (Limit DNE) Non - RemovableLHL E RHLL8a g(c)Theorema on ContinuityIff(x)andg(x)arebothcontat x = c , then f I g , fg and flgarealsocontatx=CIf f(x) is cont at x = c but g(x)is discout at x = C,- Hx) I g(x) is discont of the sametype at x=c- f(x) g(x)cannot sayIff(x)is discont at x= C but g(x)is cont 20 x = c, f(x). g(x) will bewho IF g(c) = 0 and f has finite DatePageLHL and RHL at x = CNote that -non zerocontxdiscount=discountzerocontxfinite discount=contzerocontxinfinite discoue=discovecannotIff(2)sayisdiscountatx =Canddiscont at x = d, / theng(2)is-f(x)+g(x)willbediscontatx=c,d- f(x) g(x)cannit sayIff(x)andg(x)arebothdiscountatx=C,then- f(x) + g(x)cannot say- f(x)g(x)cannot say.ContinuityofImportantFunctions-Dirichlet FunctionItisafunctiondefinedonthebasisoftheinputbeingrationalorirrationalFor example, f(x) =EQTxQall limits areoscillating sothe graph isdiscontinuouseverywherethe fn is periodic,but without aThis function CAN be contfundamental period.at the places the twoexpressions, i.e., interest -GIF and FPE[x] and {x} are discontinuousat IntegersThe possible points of discontinuity ->of [g(x)] are at g(x) E Z and g(x)dcnote If t is cont + If is also contHowever, the reverse is not guarenteed

Continuity Notes

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