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Proof of the integrability of a step function

  1. Jan 26, 2004 #1
    My second course in analysis and i have a problem which i cant understand

    Let f (x) = 1 if 2<=x<4
    2 if x =4
    -3, if 4<x<=7
    Prove that this function is integrable on [2,7], state its value and prove that it is what you say it is.
    Obviously integral of f from [2,7] is -7. but its proof and the integrability have me and my friends snagged.

    Suggestions anyone?
  2. jcsd
  3. Jan 26, 2004 #2


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    For any subdivision of [2,7] you can always choose a "finer" subdivision that includes 4 as a break point. That way you can isolate the discontinuity into two subdivisions, say [4-&delta;1,4] and [4,4+&delta;2]. What happens to the area of the rectangle based on those as &delta;1 and &delta;2 go to 0?
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