Let $L(f)$ denote the length of a curve $f$, if $f = \lim\limits_{n\to\infty} f_n$ then do we necessarily have that $L(f) = \lim\limits_{n\to\infty} L(f_n)$? I assume that we will have some continuity restrictions on $f, f_n$ but I'm not certain.
The curve I am particularly looking at is the Hilbert space-filling curve. Each iteration has length $2^n - \frac{1}{2^n}$, so can we definitely conclude from just this that the Hilbert curve itself has infinite length?
We don't have equality (e.g. consider the graphs of $\sin(2^n x)/n$, $0 \le x \le 1$ approaching a straight line), but we do have $L(f) \le \liminf L(f_n)$.