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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!

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    Constant functions are often considered to be monotone (on [Wikipedia](http://en.wikipedia.org/wiki/Monotonic_function) for example).2012-04-19
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    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

2 Answers 2

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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 $0<x\leq 1$ and $f_\gamma(x)=-5{}$ satisfy (c)

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

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    In 3. you for got a space in "_or 0<x\leq1 and_". You might want `$0\lt x\leq 1$`2012-04-19
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    @PeterT.off: There are at least continuum many. One can get more, by using the theorem that the function $f$ on say $[0,1]$ is Riemann integrable iff the set of points of discontinuity of $f$ has measure $0$.2012-04-19
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    @AndréNicolas Ok, thanks for clearing that up.2012-04-19
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    @leo thanks, I have fixed it now :)2012-04-19
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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)...

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    If you think b is monotone, so is c.2012-04-19
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    Any thoughts on a function that is not monotone but still riemann integrable?2012-04-19
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    @Mark Any non-monotone, continuous function would do, such as $f(x)=\sin(2\pi x)$. A more interesting example would be $f(x)=x\sin(1/x), x\ne 0$, $f(0)=0$.2012-04-19
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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=1/2$, let $f(1/2)=17$. – 2012-04-19
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    @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.2012-04-19
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    To find a function that is not monotone anywhere but integrable, consider the Weierstrass function.2012-06-22