[arutar]

This is a reasonable candidate for a notion of a "Fractal" but these days it is generally accepted that there is no universal definition.

For instance, this definition would not include objects such as the Devil's staircase [1] or more generally images of the unit interval under a continuous monotonically increasing function.

Some more exposition about attempts to rigorously define the notion of a fractal can be find in the introduction to Kenneth Falconer's book [2]

[1] https://en.wikipedia.org/wiki/Cantor_function

[2] https://zbmath.org/1285.28011

Edit: Fixed incorrect zbmath link.

[dbqpdb]

Another issue is that the conventional usage of the term 'fractal' implies some degree of 'self-similarity' across scale, which is a very nebulous concept and is not at all captured by defining the term with respect to sets and various measures.