Cantor function: Difference between revisions

Revision as of 23:09, 16 March 2020

Revision as of 18:24, 17 March 2020

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Then, for every integer {{nowrap|''n'' ≥ 0}}, the next function ''ƒ''_''n''+1(''x'') will be defined in terms of ''ƒ''_''n''(''x'') as follows:

Let ''ƒ''_''n''+1(''x'') = {{nowrap|0.51/2 × ''ƒ''_''n''(3''x'')}},  when {{nowrap|0 ≤ ''x'' ≤ 1/3 }};

Let ''ƒ''_''n''+1(''x'') = 0.51/2,  when {{nowrap|1/3 ≤ ''x'' ≤ 2/3 }};

Let ''ƒ''_''n''+1(''x'') = {{nowrap|0.51/2 + 0.51/2 × ''ƒ''_''n''(3 ''x'' − 2)}},  when {{nowrap|2/3 ≤ ''x'' ≤ 1}}.

The three definitions are compatible at the end-points 1/3 and 2/3, because ''ƒ''_''n''(0) = 0 and ''ƒ''_''n''(1) = 1 for every ''n'', by induction. One may check that ''ƒ''_''n'' converges pointwise to the Cantor function defined above. Furthermore, the convergence is uniform. Indeed, separating into three cases, according to the definition of ''ƒ''_''n''+1, one sees that