Ok thanks very much for this. I see now that I was being a bit silly when I suggested that sin(1/x) was probably not measurable.
Perhaps this is a better question to highlight the problems I'm having with Lebesgue integration:
Is f(x) tan x integrable over the interval (0,pi/2)?
Now we can show that f is measurable on the interval, and hence continuous. However, it is unbounded and hence not necessarily integrable. I know we can say " " and probably this to show the integral doesn't exist, but this isn't really the point. I'm after something that show from first principals or standard results (eg Monotone convergence theorem etc) why this isn't integrable over the stated interval. Thanks.