How do I show these two functions are integrable? Let $f:[a,b]\to \mathbb R $ be an integrable function. Define $$f_+(x)=\begin{cases}f(x)&\text{if } f(x)\geq0,\\\0&\text{if }f(x)<0, \end{cases} \text{ and } f_-(x)=\begin{cases}0 & \text{if }f(x)\geq0,\\\\-f(x) &\text{if }f(x)<0\. \end{cases}$$ Prove that $f_+,f_-:[a,b]\to\mathbb{R}$ are integrable and $$\int_a^bf(x)dx=\int_a^bf_+(x)dx -\int_a^bf_-(x)dx.$$ I see that $f(x)=f_+(x)-f_-(x)$, so the last part seems fairly obvious. I'm not how to show the functions are integrable, though. I feel like I must be missing something obvious, but their upper and lower sums don't seem to relate nicely to $f(x)$.

Thanks to copper.hat's remark it finally made -clonk- and here it is: $$ \DeclareMathOperator{abs}{abs} f_+(x) = \max(f(x),0) = \frac{\abs(f(x))+f(x)}{2} \\\ f_-(x) = -\min(f(x), 0) = \frac{\abs(f(x))-f(x)}{2} $$ We need the theorem that if $f$ is Riemann integrable on $[a,b]$, then so is $\abs(f)$ (Link). Together with linearity, this gives integrability of $f_+$ and $f_-$ on $[a,b]$.

Subtraction of the two equations above gives $f_+(x) - f_-(x) = f(x)$, together with the integrability and linearity it should yield the decomposition $$ \int\limits_a^b f(x) \, dx = \int\limits_a^b f_+(x) \, dx - \int\limits_a^b f_-(x) \, dx $$