Section1.8.pdf - Math 140 Fall 2016 1.8 Continuous...

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Math 140 Fall 2016§1.8 Continuous functionsThe word continuous, which means uninterrupted in time, comes from Latin. In Latin continuousmeans ”hold together”.Definition 1.Suppose a functionfis defined on an open interval that contains the pointa. Wesay thatfiscontinuous at a numberaif limxaf(x) =f(a).Note that the definition requires that the following things must be satisfies in order forfto becontinuous ata1.fmust be defined ata, that is,ais in the domain off(i.e. you are able to calculatef(a));2. limxaf(x) must exist and finally3. limxaf(x) =f(a).Ifffails any of 1.–3., then we say thatfisdiscontinuousata.Types of discontinuityThe removable discontinuity:fhas a removable discontinuity ataif limxa-f(x) =L=limxa+f(x), but the limit atais different thanf(a) (fmay not be defined ata).The jumping discontinuity:We say thatfhas a jumping discontinuity atx=aiflimxa-f(x)negationslash= limxa+f(x), but both one-sided limits exist.The infinite discontinuity

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