How to start proving |f(x)| is integrable if f(x) is?

Question

I've got the following question that I have very little idea on how to approach proving:

I was wondering if someone could send some hints/starter lines so that I actually know where this proof is headed? I'm unsure what exactly we're meant to do here :S

Answer 1

Sketch:

One of the ways that integrability of functions $f:[a,b]\rightarrow\mathbb{R}$ is determined is as follows:

Let $f^+(x)=\begin{cases}f(x)&f(x)>0\\0&f(x)\leq 0\end{cases}$ and $f^-(x)=\begin{cases}-f(x)&f(x)<0\\0&f(x)\geq 0\end{cases}$

Then, we can define $f$ to be integrable iff both $f^+$ and $f^-$ are integrable. Then we define $$ \int_a^b f(x)dx=\int_a^bf^+(x)dx-\int_a^bf^-(x)dx. $$

In the case of $|f|=f^++f^-$, we have $$ \int_a^b |f(x)|dx=\int_a^bf^+(x)dx+\int_a^bf^-(x)dx $$

From here, since $\int_a^bf^+(x)dx$ and $\int_a^bf^-(x)dx$ are both positive (as the functions are positive), you can get the necessary inequalities directly.

Answer 2

Let $\Pi$ be a partition of $[a,b]$.

It takes place that $\omega(|f|,\Pi)\leq\omega(f,\Pi),$ as for every sub-interval:

$$\omega(|f|,J)=\sup_{x,y\in J}(|f(x)|-|f(y)|)\leq\sup_J(|f(x)|-|f(y)|)=\omega(f,J)$$