I want to find the family of functions with this property:
$$\epsilon F\left(\frac{x}{\epsilon}\right)=\epsilon^r F(x)\quad\text{with}\ x,\epsilon, r \in \mathbb{R}$$ ($\epsilon$ and r are constants)
An example of those functions is $F(x)=x^{1-r}$. Is the only one I found. Some other examples of members different from this one would be useful.
An equivalent problem is to find the functions $F'{(x)}=\epsilon^r F'{(\epsilon\cdot x)}$
If the function has a Taylor series $F{(x)}=\sum_{i=0}^{\infty}a_i x^i$ then it follows $\epsilon F{(\frac{x}{\epsilon})}\,=\,\sum_{i=0}^{\infty}a_i \frac{x^i}{\epsilon^{i-1}}\,=\,\sum_{i=0}^{\infty}a_i \epsilon^r x^i$
From the definition, if $A{(x)}$ and $B{(x)}$ belong to the family, then any linear combination of A and B also belong to the family.
If $F{(x)}$ is an area, then that area has fractional dimension r according to the definition of Hausdorff dimension.