This is a basic example of a function of bounded variation on [0,1] but not continuous on [0,1].

The key Theorem regarding functions of bounded variation is Jordan’s Theorem: A function is of bounded variation on the closed bounded interval [a,b] iff it is the difference of two increasing functions on [a,b].

Consider

Both and are increasing functions on [0,1]. Thus by Jordan’s Theorem, is a function of bounded variation, but it is certainly not continuous on [0,1]!

### Like this:

Like Loading...

*Related*