Examples of bounded continuous functions which are not differentiable Most often examples given for bounded continuous functions which are not differentiable anywhere are fractals.If we include probabilistic fractals exact self-similarity is not required. Are their examples of functions which are bounded ,continuous, not differentiable anywhere and can not be modeled as fractals?

It might be worth pointing out that the typical function, in the sense of Baire category, has this property.  More specifically, Let $X=C([0,1])$ denote the set of all continuous, real valued functions defined on the unit interval endowed with the sup norm.  Let $S\subset X$ denote the set of all functions that nowhere differentiable and let $T\subset X$ denote the set of all functions whose graph has Hausdorff dimension $1$.  Then $S$ and $T$ are both residual sets - i.e., each is the complement of a countable collection of nowhere dense sets.  As a result, their intersection is second category as well and, in particular, non-empty.
The fact that $S$ is second category is a classic theorem of Stefan Banach - in fact, it's the seminal result of this type.  A proof may be found in chapter 11 of the important book Measure and Category by John Oxtoby.  The fact that $T$ is second category is proven by Humke and Petruska in Volume 14 of the The Real Analysis Exchange, though the title is "The packing dimension of a typical continuous function is 2" and it's proven in some other references as well.
