On Jun 17, 7:04 am, Paul Rubin <http://phr...@nospam.invalid> wrote: > I think a typical example of a curve that's continuous but not > uniformly continuous is > > f(t) = sin(1/t), defined when t > 0 > > It is continuous at every t>0 but wiggles violently as you get closer > to t=0. You wouldn't be able to approximate it by sampling a finite > number of points. A sequence like > > g_n(t) = sin((1+1/n)/ t) for n=1,2,... > > obviously converges to f, but not uniformly. On a closed interval, > any continuous function is uniformly continuous.
Right, but pointwise convergence doesn't imply uniform convergence even with continuous functions on a closed bounded interval. For an example, take the sequence g_n (n >= 0), of continuous real-valued functions on [0, 1] defined by: g_n(t) = nt if 0 <= t <= 1/n else 1 Then for any 0 <= t <= 1, g_n(t) -> 0 as n -> infinity. But the convergence isn't uniform: max_t(g_n(t)-0) = 1 for all n. Maybe James is thinking of the standard theorem that says that if a sequence of continuous functions on an interval converges uniformly then its limit is continuous? Mark -- http://mail.python.org/mailman/listinfo/python-list
