I have a fractal I am considering on $[0,1]$ by making "recursive tent functions".
I want to construct an explicit sequence of functions $f_n:[0,1]\rightarrow \mathbb{R}$ from the following three, I am struggling to get a closed form, despite my intuition for what the graph "should" look like. I also am interested in properties of this function.
$f_0(x)= 2x\chi_{[0,1/2)}+(-2x+2)\chi_{[1/2,1]}$ which looks like two lines linked together at $1/2$, the first piece is increasing and second is decreasing. Now we cut $f_0$ "dyadically" to obtain: $f_1(x)=2x\chi_{[0,1/4)}+(-2x+1)\chi_{[1/4,1/2)}+(2x-1)\chi_{[1/2,3/4)}+(-2x+2)\chi_{[3/4,1)}$
We repeat this again, cutting $f_1$ "dyadically" to obtain:
$f_2(x)=(2x)\chi_{[0,1/8)}+(-2x+1/2)\chi_{[1/8,1/4)}+(2x-1/2)\chi_{[1/4,3/8)}...$ and so on.
I want $f_n$, I know it will be similar to $f_n(x)=(2x)\chi_{[0,2^{-n})}+...+(-2x+2)\chi_{[\frac{2^n-1}{2^n},1]}$.
I also cautiously guess $f_n\rightarrow f=0$ uniformly (as I am pretty sure we have pointwise convergence). Assuming we have uniform convergence, this means that $\int_{0}^{1}f_n d\mu=0$ as $n\rightarrow \infty$.
Also $\forall n$ we have $f_n$ is not monotonic, yet the limit function $f=0$. $f_n$ is differentiable, as for each sub-interval, we have linear functions.
FRACTAL: 
I realize the fractal as drawn cannot be considered as a function, but perhaps can be parameterized in a clever way where I can still study some of its analytic properties.