DIMENSION ANALYSIS OF CONTINUOUS FUNCTIONS WITH UNBOUNDED VARIATION

In this paper, we mainly discuss fractal dimensions of continuous functions with unbounded variation. First, we prove that Hausdorff dimension, Packing dimension and Modified Boxcounting dimension of continuous functions containing one UV point are 1. The above conclusion still holds for continuous functions containing finite UV points. More generally, we show the result that Hausdorff dimension of continuous functions containing countable UV points is 1 also. Finally, Box dimension of continuous functions containing countable UV points has been proved to be 1 when f(x) is self-similar.