Let f: [a,b] -> R be continuous. Use Riemann's condition and uniform continuity of f to prove that f is integrable.
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1)Continous functions on closed intervals are uniformly continous.
2)Let $\displaystyle \epsilon > 0$ there is $\displaystyle \delta > 0$ so that $\displaystyle |x-y|<\delta \implies |f(x)-f(y)|<\epsilon$ for all points in interval.
3)Let $\displaystyle P=\{x_0,x_1,...x_n\}$ be partition with $\displaystyle \max (x_{k+1}-x_k) < \delta$
4)Then $\displaystyle U(f,P)-L(f,P)$ (the difference of the upper and lower sums with respect to $\displaystyle P$) can be made sufficiently small.