(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let f:[a, b][itex]\rightarrow[/itex][m, M] be a Riemann integrable function and let

[itex]\phi[/itex]:[m, M][itex]\rightarrow[/itex]R be a continuously differentable function

such that [itex]\phi[/itex]'(t) [itex]\geq[/itex]0 [itex]\forall[/itex]t (i.e. [itex]\phi[/itex]

is monotone increasing). Using only Reimann lemma, show that the composition [itex]\phi[/itex][itex]\circ[/itex]f is Riemann integrable.

2. Relevant equations

Riemann lemma - f: [a, b] [itex]\rightarrow[/itex] is Riemann integrable iff for any [itex]\epsilon[/itex]>0 [itex]\exists[/itex]a partition P such that U(P, f) - L(P, f) < [itex]\epsilon[/itex].

Function f is Riemann integrable hence it is bounded by [m, M]. Thus [itex]\forall[/itex]

x[itex]\in[/itex][a, b],lf(x)l [itex]\leq[/itex] max{m, M}.

Also, since the domain of [itex]\phi[/itex] is compact and the function is monotone and increasing, by the Extreme Value Theorem, it achieves a maximum and a minimum on [m, M], hence [itex]\phi[/itex] is also bounded. Thus, [itex]\phi[/itex]((f(a)) and [itex]\phi[/itex](f(b)) is bounded by some constant, K.

Also know since f is Riemann integrable that there exists a partition P such that

U(P, f) - L (P, f)< [itex]\epsilon[/itex]

We must show U(P,[itex]\phi[/itex](f(x))) - L(P, [itex]\phi[/itex](f(x)))<[itex]\epsilon[/itex].

I think I have most of the major pieces, can someone suggest how to put it together?

Thank you.

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# Homework Help: Show [itex]\phi[/itex][itex]\circ[/itex]f is Riemann integrable

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