The constant function c is everywhere integrable.
If f is integrable over an interval, it is integrable over every subinterval. Suppose there is a subinterval with no integral. Recall that the lower sums are always less than the upper sums, for all nets. As nets get finer, lower sums increase and upper sums decrease, yet lower sums always stay below the upper sums. Lower sums and upper sums converge over the entire interval, but not so for our pathological subinterval. Let k be a real number between the two limits; a point in the gap. Choose ε so that k±ε is still in the gap. Now everyn net, no matter how fine, produces a lower and upper sum that differ by at least 2ε. This is true for the subinterval, and it holds for the entire interval. Thus lower and upper sums don't converge, and our original integral can't exist. This is a contradiction, hence every subinterval is integrable. Using a similar argument, every subregion inside an integrable region is integrable.
Lets consider a function that is not integrable. If f() = 0 for x rational and 1 for x irrational, lower sums are 0 and upper sums are 1, hence there is no limit.