Let f:[a,b]---->R be Riemann integrable. Define F:[a,b]---->R
by
F(x)=$\displaystyle \int_a^x f(t) dt$
ii) Prove that F is bounded.
F is bounded if |F'|<=f yes? how do we know f is bounded?
thanks
Since :
$\displaystyle F(x+h) =\int^{x+h}_{a}f(x)dx =\int^{x}_{a}f(x)dx+\int^{x+h}_{x}f(x)dx =F(x)+\int^{x+h}_{x}f(x)dx$ = F(x) +hf(c) ,where x<c<x+h, by the mean value theorem of integrals.
Hence $\displaystyle lim_{h\to 0}F(x+h) = F(x)$
Hence F continuous on [a,b] ,thus bounded on [a,b]