In 1 Hans Sagan provides a parametrization for the von Koch curve and a proof for the continuity of such parametrization. The parametrization is done by dividing the triangle of vertices $\lbrace (0,0),\ (1,0),\ (1/2,\sqrt{3}/6)\rbrace$ into four squares at each iteration as is done here and providing a total order on the triangles created at a given iteration for every iteration. Then, he claims:
Now, we define the coordinates of the point on the von Koch curve that corresponds to the parameter value $t\in \mathcal{J}$ as follows: if $t$ is one of the partitition points, its image $f(t)$ shall be the corresponding nodal point. Otherwise, $t$ is uniquely determined by a sequence of nested closed intervals of lenths $1/4^n$ (the intervals of the partitions) to which, in turn, there corresponds a unique sequence of closed nested triangular regions with diameters $1/3^n$ that defines a unique point $f(t)\in E^2$. Conversely, with every point $P$ on the von Koch curve, there corresponds a unique pre-image $t\in \mathcal{J}$: if $P$ lies on the von Koch curve and is a nodal point, then its unique pre-image in $\mathcal{J}$ is the point common to the two intervals that correspond to the two triangles. Otherwise, $P$ is uniquely determined by a sequence of nested closed triangular regions that shrink into $P$ to which there corresponds a unique sequence of nested closed intervals that shrink into the unique pre-image of $P$. The mapping is continuous: if $|t_1- t_2| < 1/4^n$, then, $t_1,t_2$ lie, at worst, in two adjacent subintervals of length $1/4^n$ each and, their images $f(t_1),f(t_2)$ lie, at worst, in two adjacent triangles of diameter $1/3^n$ each. Hence, $|f(t_1) -f(t_2)| < 2/3^n$.
Could anyone please explain to me why does his reasoning proof that $f$ is continuous?
Thank you in advance for your anskwers!
REFERENCES
1 Hans Sagan (1994) The taming of a monster: a parametrization of the von Koch Curve, International Journal of Mathematical Education in Science and Technology, 25:6, 869-877, DOI: 10.1080/0020739940250612