91Ó°ÊÓ

Suppose that \(f\) is Riemann integrable on \([a, b]\) and define the function $$ F(x)=\int_{a}^{x} f(t) d t $$ (a) Show that \(F\) satisfies a Lipschitz condition on \([a, b]\); that is, that there exists \(M>0\) such that for every \(x, y \in[a, b]\), $$ |F(y)-F(x)| \leq M|y-x| $$ (b) If \(x\) is a point at which \(f\) is not continuous is it still possible that \(F^{\prime}(x)=f(x) ?\) (c) Is it possible that \(F^{\prime}(x)\) exists but is not equal to \(f(x) ?\) (d) Is it possible that \(F^{\prime}(x)\) fails to exist?

For what values of \(p\) is the integral \(\int_{1}^{\infty} x^{-p} d x\) convergent?

Show directly from the definition that the characteristic function of the rationals is not Riemann integrable but is generalized Riemann integrable on any interval and that \(\int_{0}^{1} f(x) d x=0\).

Let \(f\) be a function on an interval \([a, b]\) with the property that for every \(\varepsilon>0\) there are a pair of step functions \(L(x) \leq f(x) \leq U(x)\) so that $$ \int_{a}^{b}(U(x)-L(x)) d x<\varepsilon $$ Show that \(f\) is Riemann integrable.

Show that the integration by parts formula of extends to the case where \(f\) and \(g\) are continuous and \(f^{\prime}\) and \(g^{\prime}\) are Riemann integrable.