Let $f:[a,b]\rightarrow \mathbb{R}$ and $f(x)= \begin{cases} a_n,& \text{if }x= \frac{1}{n}, n\in\mathbb{N} \\ 0 & \text{otherwise} \end{cases}$ where $a_n$ is bounded sequence always larger than zero. Investigate the function for integrability.
I know that a function is riemann integrable if it is monotone. For this case we can choose $x_1=\frac{1}{4} , x_2=\frac{1}{\sqrt2},x_3=\frac{1}{\sqrt5},x_4=\frac{1}{2}$ to show that $x_1<x_3<x_4<x_2$ with $f(x_3)=0<f(x_1)$, $f(x_2)=0<f(x_4).$ From this it follows that $f$ isn't monotone so not necessarily integrable . The second part is where I'm having trouble, if a function is continuous on an interval it is said to be riemann integrable but considering that we've only been given that $a_n$ is bounded and nothing else I don't know how to proceed. Is the function simply discontinuous everywhere? I'd appreciate some help about this.