# Is $f(x)=\left.\begin{cases}x\,\text{sgn}(\sin\frac{1}{x})&\text{if$x\neq0$}\\0&\text{if$x=0$}\end{cases}\right\}$ Riemann integrable?

For $$x\in[-1,1]$$, let $$f(x)= \begin{cases} x\,\operatorname{sgn}(\sin\frac{1}{x}), &\text{if x\neq0} \\ 0, &\text{if x=0} \end{cases}$$ where $$\text{sgn}$$ denotes the signum function. Then:

1. $$f$$ is continuous on $$[-1,1]$$
2. $$f$$ is not differentiable at any point of $$[-1,1]$$
3. $$f$$ is Riemann integrable on $$[-1,1]$$.
4. The set of points of discontinuity of $$f$$ in $$[-1,1]$$ is finite.

I found this question in a previous year entrance paper. The answer given is option $$3$$. Now, I know that $$f$$ is discontinuous at $$x=0$$, and that $$f$$ is differentiable at many points in $$[-1,1]$$. But what about options $$3$$ and $$4$$? Aren't they both essentially the same?

• I always cringe when I see ”Reimann” integrable on this site. It is "Riemann”!!! Jul 27, 2020 at 8:12
• I wonder why this happens so often: math.stackexchange.com/search?q=Reimann Jul 27, 2020 at 8:13
• @MartinR According to "How common is your last name" the last name "Reimann" is more common than "Riemann". But I digress Jul 27, 2020 at 8:16
• The set of points of discontinuites is a countably infiinite set so 3) is true and 4) is false. Jul 27, 2020 at 8:16
• Wonder why is $f$ discontinuous at $x=0$. Jul 27, 2020 at 8:20

For each $$0 < \epsilon <1$$, $$f$$ is piecewise differentiable on $$[-1,-\epsilon] \cup [\epsilon , 1]$$ and therefore Riemann integrable on those intervals. Also $$f$$ is discontinuous at all points of the infinite set $$S=\{1/k\pi \mid k \in \mathbb Z\}$$.
Finally $$f$$ is bounded on $$[-1,1]$$. So given the above, $$f$$ is Riemann integrable on $$[-1,1]$$ and 3. is correct.
Recall that a map $$f$$ that is bounded on $$[a,b]$$ and Riemann integrable on all $$[c,b]$$ with $$a is Riemann integrable on $$[a,b]$$.
• @MartinR Right. But a function that is bounded on $[a, b]$ and Riemann integrable on all $[\epsilon, b]$ for $a< \epsilon <b$ is Riemann integrable on $[a,b]$. Jul 27, 2020 at 8:25