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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!

Martin Wanvik
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Mark
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    Constant functions are often considered to be monotone (on Wikipedia for example). – Chris Eagle Apr 19 '12 at 16:35
  • 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). – Martin Sleziak Jun 22 '12 at 06:39

2 Answers2

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In fact there are at least an uncountable number of elements in each of the three classes above.

  1. For $\alpha>0$, the class of functions $f_\alpha(x) := \begin{cases} \alpha & \text{if $x$ is rational}\\ 0 & \text{if $x$ is irrational.} \end{cases}$ satisfy (a)
  2. For $\beta>0{}$, the class of functions $f_\beta(x)=\beta x(1-x){}$ satisfy (b)
  3. For $\gamma>0$, the class of functions $f_\gamma(x)=\gamma x(1-x){}$ for $0&ltx\leq 1$ and $f_\gamma(x)=-5{}$ satisfy (c)

The domain of definition of each function above is (of course) understood to be $[0,1]$

Henrik
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(a) is absolutely correct, (b): see comments - a constant function is usually considered monotone, but not strictly so, (c): correct. Of course, you can just your example from (c) for (b)...

Johannes Kloos
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    If you think b is monotone, so is c. – Ross Millikan Apr 19 '12 at 16:45
  • Any thoughts on a function that is not monotone but still riemann integrable? – Mark Apr 19 '12 at 16:51
  • @Mark Any non-monotone, continuous function would do, such as $f(x)=\sin(2\pi x)$. A more interesting example would be $f(x)=x\sin(1/x), x\ne 0$, $f(0)=0$. – David Mitra Apr 19 '12 at 16:55
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    Many things. A familiar one is $x(1-x)$. Anything continuous on $[0,1]$ is Riemann integrable. Or can use $\sin(\pi x)$. Or $f(x)=x$ for $x \le 1/2$, $f(x)=1-x$ for $1/2<x\le 1$. For neither continuous nor monotone, "mess up" one of the above functions at say $x=1/2$, let $f(1/2)=17$. – André Nicolas Apr 19 '12 at 16:55
  • @RossMillikan: You're right, I should have read that more carefully. If Mark had written $f(1/2) = 0$, $f(x) = 2$ for $x \neq 1/2$ (which is how I interpreted this), that would have been correct. – Johannes Kloos Apr 19 '12 at 17:35
  • To find a function that is not monotone anywhere but integrable, consider the Weierstrass function. – A.S Jun 22 '12 at 05:39