Integrability of Sinusoidal Function on [-1, 1]: Finding L(f, P) and U(f, P)

[neom]

The problem states:

Decide if the following function is integrable on [-1, 1]

$$f(x)=\left\{{sin(\frac1{x^2})\;\text{if}\;x\in[-1,\;0)\cup(0,\;1]\atop a\;\text{if}\;x=0}$$

where a is the grade, from 1 to 10, you want to give the lecturer in this course

What I don't understand is how to find L(f, P) and U(f, P) since when I look at the graph of the function it oscillates a lot. So how do I choose the partition. It seems I would need an infinite partition almost to make it work. Or is there another way to do it?

Any help would be much appreciated as I am really lost on this problem.'

Edit: Sorry for the function not showing properly, don't know what I did wrong there

Should be

$$sin(\frac1{x^2})\;\text{if}\;x\in[-1,\;0)\cup(0,\;1]$$
&
$$a\;\text{if}\;x=0$$

 

[Dick]

Showing something is Riemann integrable doesn't mean you have to choose partitions. You probably have some theorems that you can use to show f(x) is Riemann integrable outside of any interval $[-\epsilon,\epsilon]$. Now show you can choose upper and lower partitions inside that interval that approach 0 as ε approaches 0.