Functions of Bounded Variation II
Let’s consider the collection of functions of bounded variation on a little more deeply. It turns out that they form a subring of the ring of all real-valued functions on . Just to be clear, the collection of all real-valued functions on an interval becomes a ring by defining addition and multiplication pointwise.
Okay, so to check that we’ve got a subring we just have to check that the sum, difference, and product of two functions of bounded variation is again of bounded variation. Let’s take and to be two functions of bounded variation on , and let be a partition of . Then we calculate
where is the least upper bound of on , and is the least upper bound of . Then we find is an upper bound for the sum over the partition. In fact, this not only shows that the product is of bounded variation, it establishes the inequality .
The proofs for the sum and difference are similar. You should be able to work them out, and to establish the inequality .
We can’t manage to get quotients of functions because we can’t generally divide functions. The denominator might be at some point, after all. But if is bounded away from — if there is an with — then is of bounded variation, and . Indeed, we can check that