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..' 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 +
11 1"
3 [2 x2dx + (I (I)dx
12
2 3 ' "
J _I = x~
( [ . 2 +.: [3 + [x J
0 2 3 8 +12 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.
 Fall '08
 Tyler/Ralston

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