Determine if the function $f:[-1,1]\to\mathbb{R}$, where $f(x) \text{ defined by} $\begin{cases} 0 & \text{if $x=\pm\frac{1}{n}, n\in\mathbb{N}$ or $x=0$} \\ 1 & \text{if $x>0$ and $x\neq\frac{1}{n}. n\in\mathbb{N}$} \\ -1 & \text{if $0>x$ and $x\neq-\frac{1}{n}, n\in\mathbb{N}$} \end{cases} is Riemann integrable or not. If it is Riemann integrable, then evaluate $$\int_{-1}^{1}f.$$ So far what I have done is:
For every $\varepsilon>0$ consider $$\left[-\frac{\varepsilon}{8},\frac{\varepsilon}{8}\right]$$
$$(M-m)\left(\frac{\varepsilon}{8}-\frac{-\varepsilon}{8}\right)=2(\varepsilon/4)=\varepsilon/2.$$
I suspect $$\int_{-1}^{1}f=0$$ (in case $f$ is integrable. Now, I need to properly show that $f$ is integrable, my intuition tells me it is. But I got stuck here. How can I proceed? Would Darboux sums be useful in this exercise? Thanks!