Fractals, when viewed as functions, are everywhere continuous and nowhere differentiable. Can this also be used as a definition for fractals?
i.e. Are all fractals everywhere continuous and nowhere differentiable? And also: Are all functions that are everywhere continuous and nowhere differentiable fractals?
On
Objects with the properties that we associate with fractality have, as you point out, been studied for more than 100 years. These include examples of the graphs of continuous, nowhere differentiable functions (fun fact: in a way that can be made precise, it is reasonable to say that the "typical" continuous function will be nowhere differentiable, thus the graphs of "most" functions will be "fractal" according to your definition) as well as uncountable, nowhere dense sets like the Cantor set.
However, the term "fractal" did not enter the lexicon until the mid- to late-1970s, when it was introduced by Mandelbrot. In his 1977 work The Fractal Geometry of Nature, Mandelbrot wrote
A fractal is by definition a set for which the Hausdorff Besicovitch dimension strictly exceeds the topological dimension. [1, p. 15]
However, this definition was quickly shown to be inadequate. For example, it excludes objects such that the graph of the Cantor function (also called the Devil's staircase), which is a rectifiable curve and therefore has Hausdorff dimension 1, matching its topological definition. It should be noted that the Devil's staircase is continuous, and differentiable almost everywhere (i.e. it is differentiable except on a set of measure zero). I suspect that most mathematicians working in this area would regard the Devil's staircase as a fractal, but neither Mandelbrot's definition nor yours would allow this object to be called fractal.
Indeed, faced with the problem of pinning down the definition of a "fractal", Mandelbrot added some discussion to an appendix to the 1982 edition of The Fractal Geometry of Nature. It is worth reading the entire discussion (it runs about half a page), but the take-away is that he concludes that it would be best
...to leave the term "fractal" without a pedantic definition, to use "fractal dimension" as a generic term applicable to all the variants in Chapter 39, and to use in each specific case whichever definition is the most appropriate. [1, p. 459]
As Emilio Novati points out, Falconer has a similar discussion. I will quote it more extensively here, as he does try to outline the properties that fractals should have (at least, to the best intuition of a mathematician who knows a lot about fractals):
In his original essay, Mandelbrot defined a fractal to be a set with Hausdorff dimension strictly greater than its topological dimension... This definition proved to be unsatisfactory in that it excluded a number of sets that clearly ought to be regarded as fractals. Various other definitions have been proposed, by they all seem to have this same drawback
My personal feeling is that the definition of 'fractal' should be regarded in the same way as the biologist regards the definition of 'life'. There is no hard and fast definition, but just a list of properties characteristic of a living thing...
When we refer to a set $F$ as a fractal, therefore, we will typicall have the following in mind.
- $F$ has a fine structure, i.e. detail on arbitrarily small scales.
- $F$ is too irregular to be described in traditional geometric language, both locally and globally.
- Often $F$ has some form of self-similarity, perhaps approximate or statistical.
- Usually, the 'fractal dimension' of $F$ (defined in some way) is greater than its topological dimension.
- In most cases of interest $F$ is defined in a very simple way, perhaps recursively.
[2, pp. xx--xxi]
Again, the term is left intentionally nebulous.
More recently, Lapidus has developed a theory of "complex dimensions" associated to set in $\mathbb{R}$ and $\mathbb{R}^n$. The basic ideas can be seen as a generalization of the Minkowski dimension, brought about by some clever number-theory-flavoured ideas involving analytic continuation of certain functions. I hesitate slightly to quote Lapidus on this, as I don't know if he has committed it to writing anywhere, but I would suspect that he would define a fractal to be as set that possesses a set of complex dimensions which includes at least one complex, non-real element. This definition seems to capture the usual suspects (such as the Devil's staircase), but the ideas are a bit esoteric, and still on the edge of research mathematics.
[1] Mandelbrot, Benoit B., The fractal geometry of nature. Rev. ed. of “Fractals”, 1977, San Francisco: W. H. Freeman and Company. 461 p. (1982). ZBL0504.28001.
[2] Falconer, Kenneth, Fractal geometry. Mathematical foundations and applications, Hoboken, NJ: John Wiley & Sons (ISBN 978-1-119-94239-9/hbk). xxx, 368 p. (2014). ZBL1285.28011.
[3] Lapidus, Michel L.; van Frankenhuijsen, Machiel, Fractal geometry, complex dimensions and zeta functions. Geometry and spectra of fractal strings, Springer Monographs in Mathematics. New York, NY: Springer (ISBN 0-387-33285-5/hbk; 0-387-35208-2/ebook). xxiv, 464 p. (2006). ZBL1119.28005.
The problem is that we do not have a ''good definition of fractal''.
In it seminal book Fractal geometry, Kennet Falconer, say :
This is a strong statement, but reflect the difficulty to comprehend on a single definition all the ''objects'' that are called ''fractals''.
It seems that the most used definition is soemthing as:
It is true that, historically, the studies of what now are called fractals, started from the study of everywhere continuous but not differentiable functions, but the Cantor set ( that is considere a classical exemple of fractal) has a characteristic function that is not continuous.