Suppose that f is continuous on 0 1 Must there be a nondegen

Suppose that f is continuous on [0, 1]. Must there be a nondegenerate closed subinterval [a, b] of [0, 1] for which the restriction of f to [a, b] is of bounded variation?

Solution

f(x) might be a constant function

It\'s true, but it doesn\'t contradict the claim, as a constant function is of bounded variation.

I think the following can give a counter-example: for each integer n, let fn be the polygonal interpolation of the points (2—k,(1)k(2—k,(1)k) and f:=k1k2fkf:=k1k2fk. This define an absolutely continuous function which is not of bounded variation on any interval [a,b].