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►For an example, see Figure 1.4.1►►►Figure 1.4.1: Piecewisecontinuous function on .
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►If is continuous or piecewisecontinuous, then
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►If is continuous or piecewisecontinuous on , then
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►If is of period , and is piecewisecontinuous, then
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►For
piecewisecontinuous on and real ,
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►If and are the Fourier coefficients of a piecewisecontinuous function on , then
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►A function is piecewisecontinuous on , where and are intervals, if it is piecewisecontinuous in for each and piecewisecontinuous in for each .
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►Sufficient conditions for the limit to exist are that is continuous, or piecewisecontinuous, on .
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►Moreover, if are finite or infinite constants and is piecewisecontinuous on the set , then
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►If is continuous and is piecewisecontinuous on , then
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►If is piecewisecontinuous, then
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►Also assume that is piecewisecontinuous on .
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►If and are piecewisecontinuous, then
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►If is piecewisecontinuous on and the integral (1.14.47) converges, then
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►Alternatively, assume is real and continuous and is piecewisecontinuous on .
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►Assume is real and continuous and is piecewisecontinuous on .
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►Assume is real and continuous and is piecewisecontinuous on .
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►More generally, assume is piecewisecontinuous (§1.4(ii)) when for any finite positive real value of , and for each , converges absolutely for all sufficiently large values of .
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