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Real analysis

  1. Nov 18, 2008 #1
    1. The problem statement, all variables and given/known data

    Let f map [a,b]-->R be a continuous nonegative function. Suppose Integral f(x)dx from a to b = 0 show that f = 0 on [a,b]


    3. The attempt at a solution

    Just not sure if this is good or not..

    so the lower sum <= 0 = integral f(x) dx

    but the lower sum must be 0 since f is non negative.

    now if f >0 then the lower sum >0 so this is a contradiction, f = 0.
     
  2. jcsd
  3. Nov 18, 2008 #2
    you need to work the fact that f is continous in there.

    If you consider a function i.e f(x)= 1 for rationals , 0 for irrationals; the integral of this is zero, the lower sum is zero, but f is in no way 0 throughout the entire interval.
     
  4. Nov 18, 2008 #3

    Dick

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    Homework Helper

    Sort of. The idea is right. But you didn't use that f is continuous, and if f isn't continuous, it's not true. Show that if f is NOT equal to 0, then that implies the integral of f is greater than zero. Pick a point x where f(x)>0. Can you see how to use continuity at x to show you can put a positive area rectangle under f (like in a lower sum)?
     
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