Examples
The function
is not of bounded variation on the interval
While it is harder to see, the continuous function
is not of bounded variation on the interval either.
At the same time, the function
is of bounded variation on the interval . However, all three functions are of bounded variation on each interval with .
The Sobolev space is a proper subset of . In fact, for each in it is possible to choose a measure (where is the Lebesgue measure on ) such that the equality
holds, since it is nothing more than the definition of weak derivative, and hence holds true. One can easily find an example of a BV function which is not : in dimension one, any step function with a non-trivial jump will do.
Read more about this topic: Bounded Variation
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