Answer to Question #310732 in Real Analysis for ABC2468

Question

Prove that if f and g are Riemann integrable on [a, b], then f · g and f + g are Riemann integrable on [a, b].

Answer 1

Let D_f, D_g be set of all discontinuations of f and g.

f+g is integrable iff,

$Df\bigcap Dg$

has a measure zero

We have,

$Df\bigcap Dg \subset$ $Df$ giving the set on left having a measure zero

Hence f+g is integrable

If f,g are both Riemann integrable then f.g is also integrable

this is proved by proving that f² is integrable

$f(g)=\frac{1}{2}((f+g)^2-f^2-g^2)$

$U(f^2,P)-L(f^2,P) =\sum(Mi^2-m i^2)\Delta xi <2T\frac{\epsilon}{2T}=\epsilon$

Here, $Sup f(x) of f^2$ is integrable and therefore f.g is integrable.