Example of a Riemann integrable function which is not a simple function

Question

I'm looking for an example of a Riemann integrable function which isn't simple?

I know that all simple functions $f: I \rightarrow E $ ( where $I \subset \mathbb{R}$ is an interval and $E$ is a Banach space) are Riemann integrable but the inclusion is strict and I don't know where to look for a not-simple but Riemann integrable function.

Could you help me?

Thank you

Answer 1

A simple function is defined as a finite linear combination of characteristic functions. However,

$$f(x)=x$$ defined on $[a,b]$ and zero otherwise, is not a characteristic function but it is Riemann integrable (in fact on all of $\mathbb{R}$).