2007-11-02 Bounded Piecewise Continuous Functions

2007-11-02 Bounded Piecewise Continuous Functions - ~ ........

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Unformatted text preview: ~ ..... ..' I}- y ."! 4 Bounded Piecewise Continuous Functions Although we are mainly interested in continous functions. many functions in applications are piecewise conrinuous. All bounded piecewise cominuous functions are integmhle (~IS many unbounded functionst arc 8.ounded on .111 interval J mC:~lI1s that for some finite constant M.llCx)1 ~ M for ,III x in J. Piecewise continuous on J means that J can be partitioned into open or h.M. open subintervuls on whkh f is continuuus. 'Ii.integr,lIe' :1hounded piecewise continuous fUIU.:tion 'h:Js a continuous extension to each that closed suhintcrv.II of the pm1iti0l1. we intcgnate the individual ext~nsiol1s and add the results. 3 2 x ..y =-1 . Piecewisecontinuous functions like this are integrated piece by piece. . IC,t) = x2. ( -I, I The Fundamental Theorem applies to boundedpiecewise continuous functions with the restriction that (d/dx) J: f(t) dt is expected to equal f (x) only at values of x at which f is continuous. :-.x. .-I:;::x<O O~.r<:! 2 < x :;::J. 3 1-.1 f(x)dx = [" (I -x)dx + 1-1 1" 3 [2 x2dx + (I (-I)dx 12 2 3 ' " J _I = x-~ ( [ . 2 +.: [3 + [-x J 0 2 3 8 --+--1--2 3 19 - 6' ...
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This note was uploaded on 09/08/2008 for the course EAC 101 taught by Professor Tyler/ralston during the Fall '08 term at University of Louisville.

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