# Rieman Integrability

Suppose $f,g:[a,b]\rightarrow\mathbb{R}$ are integrable on $[a,b]$.

Show that the function $k:[a,b]\rightarrow\mathbb{R}$ defined as $k(x)=\mathrm{max}\{f(x),g(x)\}$ is also integrable on $[a,b]$.

 $max(f,g)=\frac{f+g+|f-g|}{2}$
Another way to see that the absolute value is integrable is by using Riemann’s criteria of integrability. If f is integrable, then its set of descontinuity points has measure zero, so does $|f|$, so $|f|$ is Riemann integrable.