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

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

$$f(x)=x$$ defined on $[a,b]$ and zero otherwise, is not a characteristic function but it is Riemann integrable (in fact on all of $\mathbb{R}$).
• $\int_{-\infty}^\infty f dx = \int_a^b x dx= \frac{x^2}{2}|_a^b = .5 (b^2-a^2)$ Apr 9, 2014 at 15:32
• Here $E=\mathbb{R}$ which is clearly a Banach because the completeness of the real numbers (from undergraduate analysis) and the norm is the usual absolute value. $I=[a,b]$ here. Apr 9, 2014 at 15:35