Let $f:[a,b]\to \mathbb{R}$ is integrable on $[a,b]$. Show that $f_1:[a,b]\to \mathbb{R}$ such that $f_1(x)=\sup\{f(x), 0\}$, is integrable.

Question

Let $f:[a,b]\to \mathbb{R}$ is Riemann integrable on $[a,b]$. Show that $f_1:[a,b]\to \mathbb{R}$ such that $f_1(x)=\sup\{f(x), 0\}$, $x\in [a,b]$ is Riemann integrable on $[a,b]$.

Please give me the idea how to proceed. I have no idea to solve. Please help. Don't know what is the meaning of $\sup\{f(x),0\}$?

Answer 1

For any real numbers $s,t$, we have $$\sup\{s,t\}=\frac{1}{2}(s+t)+\frac{1}{2}|s-t|.$$ Thus $$\sup\{f(x),0\}=\frac{1}{2}f(x)+\frac{1}{2}|f(x)|.$$ Then the integrability of $\sup\{f(x),0\}$ follows from the integrability of $|f(x)|$ and $f(x)$.