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:where ''f'' is Thomae's function and 2<sup>''q''</sup> is any power of 2 that is larger than ''n'', so that ''n''/2<sup>''q''</sup> is between 0 and 1. Dunham uses "ruler function" as a synonym for Thomae's function in [http://books.google.co.uk/books?id=QnXSqvTiEjYC&pg=PA175&dq=%22ruler+function%22+Thomae&sig=ACfU3U3B0h1K-42ZyOPlR3f7KbY_QZzZHQ ''The Calculus Gallery''], as does Burn in [http://books.google.co.uk/books?id=1nteiLydVpQC&pg=PA158&lpg=PA158&dq=%22ruler+function%22+Thomae&source=web&ots=LEY1ak2-qV&sig=1rSU6rtbebN1Mxs4ZxecMeqLt6I&hl=en&sa=X&oi=book_result&resnum=12&ct=result#PPA158,M1 ''Numbers and Functions'']. We could obviously extend the article to include the MathWorld/OEIS definition of ruler function as an alternative definition. [[User:Gandalf61|Gandalf61]] ([[User talk:Gandalf61|talk]]) 10:38, 24 August 2008 (UTC)
::While that is clever, the contortions needed to establish a link between Thomae's function and the ruler function as defined in MathWorld (the 2-adic valuation of positive integers) are such that it is hard to consider one as a transformation of the other. Note that you need to modify ''n'' in a way that depends essentially on the value of ''n'' itself before feeding it into ''f'', and then need to tweek the value of ''f'' similarly; if one is going to allow that kind of things, one could "transform" almost any function into any other function. The fact is the function have vastly different properties (to name a few, Thomae's function is defined on an interval, takes arbitrarily small positive values, and is bounded above by 1; the ruler function is defined on positive integers only, has natural number values values and is unbounded). I don't think the graph of Thomae's function even vaguely resembles any ruler in existence (nor does it resemble popcorn, raindrops, or stars over Babylon for that matter). It is undeniable that some authors use "ruler function" as a name for Thomae's function, but other use it to designate 2-adic valuation, for instance in [[Concrete Mathematics]], p 113. In any case the current situation with a separate section in this article describing a ''different'' function is not really acceptable. [[User:Marc van Leeuwen|Marc van Leeuwen]] ([[User talk:Marc van Leeuwen|talk]]) 18:06, 9 April 2017 (UTC)
== Formal proof of continuity ==
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