Riemann Integrability

[O_o]

Homework Statement

Prove or give a counter example of the following statement:

If $f: [a,b] \to [c,d]$ is linear and $g:[c,d] \to \mathbb{R}$ is Riemann integrable then $g \circ f$ is Riemann integrable

Homework Equations

The Attempt at a Solution

I'm going to attempt to prove the statement is true.

Let $f(x) = ax + b$. I'm going to assume $a > 0$ and do a similar proof for $a < 0$ if this one is alright.

Fix $n \in \mathbb{N}$ such that $\frac{1}{n} \leq a$.

Fix $\epsilon > 0$

Since g is Riemann integrable $\exists P = \{y_0, y_1, ..., y_n\}$ such that $U(g,P) - L(g,P) \lt \frac{\epsilon}{n}$. Where $P$ is a partition of $[c,d]$.

Let $Q = \{x_0, x_1, ..., x_n\}$ where $x_i = \frac{y_i - b}{a}$. Since $f(x)$ is strictly increasing, $x \in [x_{i-1}, x_i] \implies f(x) \in [y_{i-1}, y_i]$.

This means $M_i = \sup\{ g(y): y \in [y_{i-1}, y_i]\} = \sup\{g(f(x)): x \in [x_{i-1}, x_i]\}$ and likewise for the infimum over the interval, which I will label $m_i$.
$$\implies \\ U(g \circ f ,Q) - L(g \circ f ,Q) = \sum_{i=1}^n \left(M_i - m_i\right) (x_i - x_{i-1}) \\ = \sum_{i=1}^n \left(M_i - m_i\right) \frac{(y_i - y_{i-1})}{a} \\ \leq \sum_{i=1}^n \left(M_i - m_i\right) (y_i - y_{i-1}) n \\ = n \left(U(g,P) - L(g,P)\right) \\ \leq n \frac{\epsilon}{n} = \epsilon$$

I was hoping someone could confirm my reasoning is okay or point out a place I made a mistake. Thanks

 

[RUber]

Looks like a solid argument to me.

 

[O_o]

Cheers. Thanks for taking the time to read it over.