1. Riemann integrals

Let f and g be bounded functions defined on [a,b].

Suppose that f is equal to zero on [a,b] except for one point. Prove that f is Riemann integrable on [a,b] and that the integral of f from a to b equals zero.

Prove the same thing except for f is equal to zero on [a,b] except for a finite number of points.

2. Originally Posted by friday616
Let f and g be bounded functions defined on [a,b].

Suppose that f is equal to zero on [a,b] except for one point. Prove that f is Riemann integrable on [a,b] and that the integral of f from a to b equals zero.

Prove the same thing except for f is equal to zero on [a,b] except for a finite number of points.
It would be extremely helpful if you could indicate where and why you are having trouble. The operative thing to notice here is that the infimum on any subinterval will trivially be zero (why?). Also, since there are finitely many non-zero points it is quite easy to construct a partition that the supremum will be zero on most intervals and less than or equal to some value on finitely man intervals (what are these intervals and what values are they less than?). So to show this set of functions is integrable we must merely pick a partition such that....what?