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?
 A: First, you have to define what you mean by a "fractal". There is only one mathematica definition of a fractal curve that I know, it is due to Mandelbrot (I think). A curve is called fractal if its Hausdorff dimension is $>1$. 
Now, back to your question. The condition of being bounded is not particularly relevant, as you can restrict any continuous function $f: R\to R$ without 1-sided derivatives to the interval $[0,1]$ and then extend the restriction to a periodic function $g$, $g(x+n)=g(x)$ for all $x\in [0,1]$, $n\in {\mathbb N}$. 
Now, take the Takagi function: it has no 1-sided derivatives at any point, is continuous and its graph has Hausdorff dimension 1 (see here). 
Edit: Note that Takagi's function does have periodic extension since $f(0)=f(1)$. For a general nowhere differentiable function $f$ you note that it cannot be monotonic (if it is nowhere differentiable). Then find $a<b$ such that $f(a)=f(b)$ and then extend to $R$ periodically using $[a,b]$ as the period.   
A: 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.
