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Rieman Integrability

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Rieman Integrability

Suppose f,g:[a,b]normal-:fgnormal-→abf,g:[a,b]\rightarrow\mathbb{R} are integrable on [a,b]ab[a,b].

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

Please help. Thank you


Observe that

max(f,g)=f+g+|f-g|2maxfgfgfg2max(f,g)=\frac{f+g+|f-g|}{2}

As f and g are Riemann integrable, it is easy now, if you already now that sums of integrable functions are integrable as well as the absolute value of integrable functions, and division (which is trivial). The best way, I think, of proving that the abolute value is integrable is using the definition of Darboux integrable, which is equivalent to Riemann integral, but is a little simpler I think.

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|f|f|, so |f|f|f| is Riemann integrable.

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