Proving integrability of a composition of functions

[bobbyjrock]

Homework Statement

Show that if f is an integrable function on [a,b] then g(x) which is defined to be sin(f(x)) is also integrable

Homework Equations

The Attempt at a Solution

I started off by trying to show that since f is integrable it has an Upper sum and a Lower sum where U(f,P)-L(f,P) < [itex]\epsilon[/itex] and then that if you take the sin of Mj and mj that since sin is continuous Mj-mj will also be less than some epsilon. I'm not sure if this works or how to go about it completely. Thanks in advance

 

[mighty2000]

for any help.

First, let's define what it means for a function to be integrable on [a,b]. A function f is said to be integrable on [a,b] if for any given ɛ > 0, there exists a partition P of [a,b] such that the difference between the upper sum U(f,P) and lower sum L(f,P) is less than ɛ.

Now, let's consider the function g(x) = sin(f(x)). We want to show that g(x) is also integrable on [a,b].

First, we need to show that g(x) is bounded on [a,b]. Since f is integrable on [a,b], it is also bounded on [a,b]. This means that there exists some constant M > 0 such that |f(x)| ≤ M for all x ∈ [a,b]. Since sin(x) is also bounded between -1 and 1 for all x, we can say that |sin(f(x))| ≤ 1 for all x ∈ [a,b]. Therefore, g(x) is also bounded on [a,b].

Next, we need to show that g(x) is also integrable on [a,b]. Let ɛ > 0 be given. Since f is integrable on [a,b], there exists a partition P of [a,b] such that U(f,P) - L(f,P) < ɛ. Now, consider the partition P' of [a,b] where each subinterval of P' is the same as the corresponding subinterval in P, but with the endpoints shifted by a small amount. More specifically, let P' = {a = x0 < x1 < ... < xn-1 < xn = b} where xi = xi-1 + δ for some δ > 0. This means that each subinterval in P' is a small enough interval such that it is contained within a subinterval in P.

Now, let's consider the upper and lower sums for g(x) on P'. We have:

U(g,P') = ∑(sin(Mj)Δxj) where Mj is the supremum of sin(f(x)) on the jth subinterval of P' and Δxj is the length of the jth subinterval of P'.

L(g,P') = ∑(sin(mj)Δxj) where mj is the inf