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


1 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 $\mathbb{R}$).

  • 1
    $\begingroup$ $\int_{-\infty}^\infty f dx = \int_a^b x dx= \frac{x^2}{2}|_a^b = .5 (b^2-a^2)$ $\endgroup$
    – Squirtle
    Apr 9, 2014 at 15:32
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    $\begingroup$ 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. $\endgroup$
    – Squirtle
    Apr 9, 2014 at 15:35
  • $\begingroup$ Thank you. I should've thought of that :) $\endgroup$
    – Hagrid
    Apr 9, 2014 at 15:36

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