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?

  • $\begingroup$ What do you mean by $\lim_n f_n = f$? That is the crucial part... $\endgroup$ – copper.hat Dec 19 '14 at 20:13
  • $\begingroup$ That $f_n$ is a sequence of functions converging to $f$. $\endgroup$ – mayhemmelody Dec 19 '14 at 20:14
  • $\begingroup$ I understand that. But there are many different kinds of convergence. Back to my first comment... $\endgroup$ – copper.hat Dec 19 '14 at 20:17
  • $\begingroup$ For the sake of this question, say it is uniformly continuous. $\endgroup$ – mayhemmelody Dec 19 '14 at 20:18
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    $\begingroup$ What is uniformly continuous? $\endgroup$ – copper.hat Dec 19 '14 at 20:20

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 $L(f) \le \liminf L(f_n)$.

  • $\begingroup$ Thank you for the counterexample. $\endgroup$ – mayhemmelody Dec 19 '14 at 20:44

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