「cantor関数」の編集履歴（バックアップ）一覧に戻る
cantor関数 - (2009/07/23 (木) 18:02:13) のソース
<math>f_0(x) := x \mbox{ for all } x \in [0,1]</math>
<math>f_1(x) := \begin{cases} \frac{3}{2}x & 0 \leq x \leq \frac{1}{3} \\ \frac{1}{2} & \frac{1}{3} < x < \frac{2}{3} \\ \frac{3}{2}x-\frac{1}{2} & \frac{2}{3} \leq x \leq 1 \end{cases}</math>
<math>f_2(x) := \begin{cases} \frac{1}{2}f_1(3x) & 0 \leq x \leq \frac{1}{3} \\ f_1(x) & \frac{1}{3} < x < \frac{2}{3} \\ \frac{1}{2}f_1(3x-2)+\frac{1}{2} & \frac{2}{3} \leq x \leq 1 \end{cases}</math>
…
<math>f(x) := \lim_{n \to \infty} f_n(x) \mbox{ uniformly on } [0,1]</math>
is a continuous, increasing function on [0,1]
with f(0)=0 and f(1)=1 that satisfies f'(x)=0 for all x in the open set [0,1]-C
where, C⊂[0,1] is the Cantor Set.
