Is it true that if a non-negative function is Riemann integrable, there must exist an upper sum U(f)?

Question

I know that if the upper sum of f, $U(f)$ equals the lower sum of f on an interval $[a,b]$ then f is Riemann integrable on [a,b] with $U(f)=L(f)= \int_{a}^{b} f(x) \,dx.$

Is the reverse true? That is if f is riemann integrable is it the case that there must exists an U(f) and L(f) with $U(f)=L(f)= \int_{a}^{b} f(x) \,dx.$?

Edit: My books gives the following:

A bounded function $f$ defined on the interval [a,b] is Riemann integrable if $U(f) = L(f).$ In this case we define $\int_{a}^{b} f$ or $\int_{a}^{b} f(x)$ to be $\int_{a}^{b} f = U(f) = L(f).$

They don't use "if and only if" so i was note sure whether it was the case that if f is riemann integrable then there must exists an U(f) and L(f) with $U(f)=L(f)= \int_{a}^{b} f(x) \,dx.$?

Answer 1

When a book uses the phrasing, "We say that an A has property P if condition C holds", they mean "exactly if" or "if and only if". Just using the word "if" is a universal shorthand for mathematical definitions.

Answer 2

By definition $U(f)$ always exist to be the infimum of upper sums over partitions. So any bounded function (which includes Riemann integrable functions) would have an upper sum.