Riemann-integration problem

Question

Here is the exercise

Let $f:[a,b]\rightarrow \mathbb{R}$ be Riemann-integrable. Prove that $f^+$, $f^-$ and $|f|$ are also Riemann-integrable, when

$$f^+=\begin{cases} f(x) & f(x)\geq 0 \\ 0 & otherwise \end{cases}$$

$$f^-=\begin{cases} -f(x) & f(x)\leq 0 \\ 0 & otherwise \end{cases}$$

This problem seems so obvious. Why wouldn't $f^+$ be integrable? Anyway, I need to prove this using this hint:

"$f$ is Riemann-integrable if with every $\epsilon>0$ there exists step functions $h\leq f\leq g $ such that $\int h -\int g <\epsilon$."

I have no idea where to start...

Answer 1

We prove the Riemann integrability of $f^+$. A similar proof can be done for $f^-$.

As $f$ is Riemann-integrable, for all $\epsilon>0$ there exists step functions $h \leq f\leq g $ such that $\int h -\int g <\epsilon$.

Now define $h^+ = \max(h,0)$ and $g^+ = \max(g,0)$. You can verify that:

1. $h^+, g^+$ are step functions.
2. You have $h^+ \le f^+ \le g^+$ for all $x \in [a,b]$.
3. And also $g^+ - h^+ \le g-h$ for all $x \in [a,b]$. This implies $0 \le \int (g^+-h^+) \le \int (g-h) \le \epsilon$

And concludes the proof.