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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]].
The ''Rvachev up function'' is closely related: {{math|up(''x'') {{=}} F(1 - {{!}}''x''{{!}}) for {{!}}''x''{{!}} ≤ 1}}.
==Values==
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