Riemann's proof of the existence of definite integrals

  • #1
Hello,

Since it was mentioned in my textbook, I've been trying to find Riemann's proof of the existence of definite integrals (that is, the proof of the theorem stating that all continuous functions are integrable). If anyone knows where to find it or could point me in the right direction, I would much appreciate it :)
 

Answers and Replies

  • #2
535
72
No, it is not true that all continuous functions are integrable (##f = 1## is continuous and not integrable on ##\mathbb{R}_+##). However, it is true that continuous functions on a segment are integrable (even piecewise continuous functions).
This comes from the fact that for every piecewise continuous functions ##f: [a,b] \to E ##, ##E## finite dimentional vector space, such as ##\mathbb{R}## or ##\mathbb{C}##, there exists a sequence ##(s_n)_n## of step functions that converges uniformely toward ##f##. Therefore the sequence ##(\int_{[a,b]} s_n)_n ## converges in ##E## as it is a Cauchy sequence, and its limit is called integral of ##f## on ##[a,b]##.
 

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