The idea is to have iterative fractals of continuous functions (see also IFS or iterated function system). In the limit, it's hard to tell whether such a fractal would be continuous by the end of things, but at the beginning it looks possible. Below is an example of a non-continuous function (Koch curve):

Base (iteration 0):
```
_____________________

Generator (iteration 1):

/\
/  \
_______/    \________

Iteration 2:

__/\__
\    /
___/\___/    \___/\___

```
Okay, so it's pretty ugly. I hope the general idea is clear however. My idea is to take some continuous ("nice") function and iterate it in a similar manner; examples of "nice" functions might include sin, a binomial with rotational symmetry about the origin, or other functions with such rotational symmetry.
```Here's one function I'm thinking of (sin(pi*(x+1))):

Base (iteration 0):

----------------------------------------------------------------------------------------------------

Generator (iteration 1):

,,oO```Oo,,
,oO`         `Oo,
,/`               `\,
,/`                   `\,
,/`                       `\,
/`                           `\
,/                               \,
/`                                 `\
,/                                     \,
/`                                       `\
,/                                           \,
,/`                                             `\,
`\                                           /`
\,                                       ,/
`\                                     /`
\,                                 ,/
`\                               /`
\,                           ,/
`\,                       ,/`
`\,                   ,/`
`\,               ,/`
`Oo,         ,oO`
``Oo,,,oO``

Iteration 2:

,.ooOO''OOo.,,
/'`         `''OOo.,
/                  `'\,
|`                     `\,
|,                       \
\                       '|
|,                      ,|
\                      /
\                    |'
\,                  /
`\               ,/                                     ,,.ooOO''````'\,
\,            /`                             ,,.ooOO'``              |
----------`\---------,/----------------------,,.ooOO'``----------------------/`--------\,-----------
|              ,,.ooOO'``                             ,/           `\
`\.,,,,..ooOO'``                                     /`              \,
,/                 `\
,|                    \
/                      \
|`                      `|
|,                       \
\                       `|
\,                     ,|
`\.,                  /
`'OOo..,         ,./
``'OOo..oOOO'`
```
The generalized form of iterations (as I see them) are as follows:

Variables:

t (for Theta. The equations below are generally considered parametric. The range for t will probably be -q to q where q is an "interesting" solution of the generative function.)
x (the infamous variable from algebra. I'll be using x for graphing as well.)
y (the companion variable for x. Ditto the graphing bit.)
p (not quite sure where I'm getting the label from. This is the number of subsections to divide the prior iteration into when iterating.)
Functions (grouped according to iteration):

Base (iteration 0):

f
0(t)=t

g
0(t)=0

Generator (iteration 1):

f
1(t)=t

g
1(t)=(pick an "interesting" function)

Iterations (iteration a):

Let F(t) be the derivative of f
a-1(t).

Let G(t) be the derivative of g
a-1(t).

Let s be the integral from -q/(p^(a-1)) to q/(p^(a-1)) of sqrt(F(t)+G(t)).

f
a(t)=fa-1(t)-sqrt(G(t)/(1+G(t)))*ga-1((p^(a-1))*s*t)/((p^(a-1))*s)

g
a(t)=ga-1(t)+sqrt(1/(1+G(t)))*ga-1((p^(a-1))*s*t)/((p^(a-1))*s)

The difficulty I keep having is the integral ("s" above). That is a basic length integral for parametric equations instead of standalone equations. Specifically, sin(x) is nasty in this integral. In graphing the fractal, let x=fa(t), and y=ga(t).

It appears that sin(x) would be quite a fascinating function to use here, but for that length integral. x^3-x is also interesting, but also difficult.