Multiplication and sum of Riemann integrable functions

Question

Consider, that $g(x)$ and $f(x)$ two Riemann integrable functions. There is $s(x) = g(x)*f(x)$. Will $s(x)$ be also Riemann integrable? The same question about sum $t(x) = g(x)+f(x)$. It seems to me, that neither $s(x)$ nor $t(x)$ are Riemann integrable, but is there some information about this?

Answer 1

The sum of two Riemann integrable functions is Riemann integrable if you consider the definition of Riemann integrable. Try to prove that $L(g + f, P) = L(g, P) + L(f, P)$ and $U(g + f, P) = U(g, P) + U(f, P)$ for any partition $P$.

$s(x)$ is in fact Riemann integrable as well. Proving it is a little more tricky. It can be seen easily by using Lebesgue Criterion (Riemann integrable functions are those that are continuous almost everywhere).