(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose f is continuous on [a,b] with f(x) >= 0 for all x in [a,b].

If [tex] \int_a^b \! f = 0 [/tex] , prove that f(x)=0 for all x in [a,b].

2. Relevant equations

3. The attempt at a solution

Assume [tex] \int_a^b \! f = 0 [/tex].

Then the lower and upper sums equal 0 also. I looked at the definitions of these and came up with,

[tex] \sum_{i=1}^n ( inf\{ f(x) : x_{i-1} \leq x \leq x_i \} )( \bigtriangleup x_i )=0 [/tex].

This looks obvious to me but that usually means I'm missing something because I don't know where to use that f is continuous.

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# Homework Help: If integral f =0, prove that f(x)=0 for all x in [a,b]

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