91Ó°ÊÓ

Let \(f(x)=x\) for rational \(x\) and \(f(x)=0\) for irrational \(x\) (a) Calculate the upper and lower Darboux integrals for \(f\) on the interval \([0, b]\) (b) Is \(f\) integrable on \([0, b] ?\)

Let \(f\) be a continuous function on \((a, b)\) such that \(f(x) \geq 0\) for all \(x \in(a, b) ; a\) can be \(-\infty, b\) can be \(+\infty .\) Show that the improper integral \(\int_{a}^{b} f(x) d x\) exists and equals $$ \sup \left\\{\int_{c}^{d} f(x) d x:[c, d] \subseteq(a, b)\right\\} $$

Let \(f\) be a bounded function on \([a, b] .\) Suppose there exist sequences \(\left(U_{n}\right)\) and \(\left(L_{n}\right)\) of upper and lower Darboux sums for \(f\) such that \(\lim \left(U_{n}-L_{n}\right)=0 .\) Show \(f\) is integrable and \(\int_{a}^{b} f=\lim U_{n}=\lim L_{n}\)

Prove the following comparison tests. Let \(f\) and \(g\) be continuous functions on \((a, b)\) such that \(0 \leq f(x) \leq g(x)\) for all \(x\) in \((a, b) ; a\) can be \(-\infty, b\) can be \(+\infty\) (a) If \(\int_{a}^{b} g(x) d x<\infty,\) then \(\int_{a}^{b} f(x) d x<\infty\) (b) If \(\int_{a}^{b} f(x) d x=+\infty,\) then \(\int_{a}^{b} g(x) d x=+\infty\)

Suppose that \(f\) is a continuous function on \([a, b]\) and that \(f(x) \geq 0\) for all \(x \in[a, b] .\) Show that if \(\int_{a}^{b} f(x) d x=0,\) then \(f(x)=0\) for all \(x\) in \([a, b]\)

Let \(f(x)=x \operatorname{sgn}\left(\sin \frac{1}{x}\right)\) for \(x \neq 0\) and \(f(0)=0\) (a) Show that \(f\) is not piecewise continuous on [-1,1] (b) Show that \(f\) is not piecewise monotonic on [-1,1] (c) Show that \(f\) is integrable on [-1,1]

Show that if \(f\) is a continuous real-valued function on \([a, b]\) satisfying \(\int_{a}^{b} f(x) g(x) d x=0\) for every continuous function \(g\) on \([a, b]\) then \(f(x)=0\) for all \(x\) in \([a, b]\).

Suppose \(f\) and \(g\) are continuous functions on \([a, b]\) such that \(\int_{a}^{b} f=\) \(\int_{a}^{b} g .\) Prove that there exists \(x\) in \([a, b]\) such that \(f(x)=g(x)\)