Verified answer

Your question as phrased doesn't make sense. Is S a finite subset of [a,b] at

which f is either undefined or not continous?

What do you mean by f(x)="i"? Does f(x)=i ( a constant)for all x in [a,b] in which

case the Riemann sums trivially converge?

If you mean f(x)=0 on [a,b]-S with a<=s1<s2<...<sk<=b then, on each subinterval [s(i),s(i+1)] f is a constant (0) and hence f is integrable on each subinterval (the Riemann sums trivially converge to 0).

Since the sum of the integrals over each subinterval is the integral (or the limit of the Riemann sum) over [a,b] it follows that f is integrable on [a,b].

Recall that how f is defined at the endpoints of any subinterval does not influence the Riemann sums.

Hope that helps a bit.