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Give an example of a function $f:[0,1] \rightarrow \mathbb{R}$ such that...

(a) $f$ is bounded, but not Riemann integrable on $[0,1]$. $$ f(x) := \begin{cases} 2x & \text{if $x$ is rational}\\ x & \text{if $x$ is irrational.} \end{cases} $$ (b) $f$ is Riemann integrable on $[0,1]$ but not monotone.

$$f(x) := 2$$

(c) $f$ is Riemann integrable on $[0,1]$ but neither continuous nor monotone.

$$f(x) := \begin{cases} 0 & \text{if $x$ is $0$}\\ 2 & \text{otherwise.} \end{cases} $$


Is this correct? Thanks!

  • 5
    Constant functions are often considered to be monotone (on [Wikipedia](http://en.wikipedia.org/wiki/Monotonic_function) for example).2012-04-19
  • 0
    If $f$ is Riemann integrable and you change the value in finitely many points, the result will be again Riemann integrable. If you known this fact, you should be able to find some examples for (b) and (c).2012-06-22

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