Limit of polygonal arcs Given a sequence of polygonal arcs $(f_n)_{n\in\mathbb{N}}$ that has limit $f$, is $f$ continuous? 
I believe it should be true as I cannot think of a counter example but not sure how to prove it. 
Thank you for your help.

Let $f_1$ be the polygonal arc defined thus.  Let $P_0=(1,0)$, $P_1=(2/3,1)$,
$P_2=(1/3,0)$, $P_\infty=(0,0)$.  Join $P_0$ to $P_1$ to $P_2$ to $P_\infty$ by straight line segments.
Let $f_2$ be the polygonal arc defined thus.  Let $P_3=(2/9,1)$, $P_4=(1/9,0)$.  Join $P_0$ to $P_1$ to $P_2$ to $P_3$ to $P_4$  to $P_\infty$ by straight line segments.
In general, for $k \ge 1$, let $P_{2k-1} =(2/3^k,1)$ and $P_{2k}=(1/3^k,0)$,
and let $f_n$ be the polygonal arc obtained by joining $P_i$ to $P_{i+1}$, for $i=0,1,\dots, 2n-1$, and finally $P_{2n}$ to $P_\infty$, by straight line segments.
We have imitated the behaviour of $\sin^2(1/x)$. The limiting curve is not continuous at $(0,0)$.   If an endpoint difficulty is not acceptable, we can modify the definitions by adding to each polygonal arc $f_n$ its reflection across the $y$-axis.
