> I still wonder sometimes whether an infinite composition of continuous functions has to be continuous (limit of a sequence) ...
No, as you expected. Take f(x) = x^2 on [0,1]. Then f(f(f(...f(x))) converges to g(x) = { 0, if 0 <= x < 1; 1, if x=1 }
No, as you expected. Take f(x) = x^2 on [0,1]. Then f(f(f(...f(x))) converges to g(x) = { 0, if 0 <= x < 1; 1, if x=1 }
Consider a pencil that is placed almost vertically, and then later drops to a horizontal position. There is a period where it is "unsure" where to go, and the more perfectly balanced it starts out, the longer this period is (assuming no wind etc.)
That's what is being discussed.