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Citation bot (talk | contribs) Alter: template type. Add: s2cid. | Use this bot. Report bugs. | Suggested by AManWithNoPlan | #UCB_webform 950/1776 |
clarify 'the same equation' |
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This function satisfies the initial condition <math>f(0) = 0</math>, the symmetry condition <math>f(1-x) = 1 - f(x)</math> for <math>0 \le x \le 1,</math> and the [[functional differential equation]] <math>f'(x) = 2 f(2 x)</math> for <math>0 \le x \le 1/2.</math> It follows that <math>f(x)</math> is monotone increasing for <math>0 \le x \le 1,</math> with <math>f(1/2)=1/2</math> and <math>f(1)=1.</math>
There is a unique extension of {{mvar|f}} to the real numbers that satisfies the same differential equation for all ''x''. This extension can be defined by {{math|1=''f''{{hsp}}(''x'') = 0}} for {{math|''x'' ≤ 0}}, {{math|1=''f''{{hsp}}(''x'' + 1) = 1 − ''f''{{hsp}}(''x'')}} for {{math|0 ≤ ''x'' ≤ 1}}, and {{math|1=''f''{{hsp}}(''x'' + 2<sup>''r''</sup>) = −''f''{{hsp}}(''x'')}} for {{math|0 ≤ ''x'' ≤ 2<sup>''r''</sup>}} with {{mvar|r}} a positive integer. The sequence of intervals within which this function is positive or negative follows the same pattern as the [[Thue–Morse sequence]].
==Values==
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