# Talk:Thomae's function

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## "Thomae's function" vs "popcorn function"

I don't have enough analysis books to do a comprehensive count, but in every book I have, this is called Thomae's Function. I know it isn't that big a deal, but "popcorn function" is so terribly informal, whereas "Thomae's function" falls in line with so many other functions whose names are called "[name]'s function" or "[name] function" or similar. Look at List of mathematical functions and tell me how many are named after people vs how many are named after food that they (apparently) resemble. Even the closest relative to this function is named properly, as the Dirichlet function (although it is in nowhere continuous function due to a merge of some sort). That section in the list includes several functions of this exact type (canonical examples/counterexamples in elementary analysis), all of which are named after people. So anyway, sorry about rambling a bit, I propose that we move it to Thomae's function, and redirect this page there. I just feel like maybe next we'll move Error function to ski-slope function (yes, I know, I'm being sarcastic). Any opinions? --Cheeser1 05:01, 27 July 2007 (UTC)

OK, I did the move. Oleg Alexandrov (talk) 02:05, 24 August 2007 (UTC)

## f(0)?

What is the value of f at 0? I would assume it to be 0 but I have no reference for this. --89.12.119.20 18:58, 12 August 2007 (UTC)

Notice that 0 is rational, and in least terms, 0=0/1. Least terms are ensured by the stipulation that gcd(p,q)=1. Thus f(0) = 1. If f(0)=0, then the function would be continuous there - this would violate the conclusion that f is discontinuous at rational points. --Cheeser1 21:16, 12 August 2007 (UTC)
Actually, by that reasoning f(0) = 1 which doesn't contradict discontinuity at all. Olaf Davis (talk) 23:46, 19 June 2009 (UTC)

Not only is f(0)=1, but f(z)=1, for any integer z.
Why? 0= 0/1 in lowest terms(intuitively), and Thomae only uses the denominator, f(0)=1/1 Similarly, z/1 -> f(z)=1/1 Nickalh50 (talk) 19:28, 14 March 2011 (UTC)

## Newer image with higher resolution

I uploaded a newer version with a higher resolution (Image:Popcorn function plot bars.png, even a SVG version up to denomiator 750 (Image:Popcorn function plot bars.svg), but this is not working at the moment, see commons:Commons:Help_desk#SVG_too_big.2C_.svg.gz__or_.svgz_was_not_accepted. What do you think? --84.72.190.27 (talk) 10:01, 27 November 2007 (UTC)

## Ruler function??

I suspect this is wrong, in what way is this a modification of the ruler function?

According to MathWorld the ruler function is The exponent of the largest power of 2 which divides a given number 2n.

It looks like this

-- teadrinker (talk) 08:53, 24 August 2008 (UTC)

The relation is that if p is the exponent of the largest power of 2 that divides n then
${\displaystyle p=\log _{2}\left(f\left({\frac {n}{2^{q}}}\right)\right)+q}$
where f is Thomae's function and 2q is any power of 2 that is larger than n, so that n/2q is between 0 and 1. Dunham uses "ruler function" as a synonym for Thomae's function in The Calculus Gallery, as does Burn in Numbers and Functions. We could obviously extend the article to include the MathWorld/OEIS definition of ruler function as an alternative definition. 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. Marc van Leeuwen (talk) 18:06, 9 April 2017 (UTC)

## Formal proof of continuity

Would it make much sense to include a formal proof of the evaluation of continuity for this article? The informal proof seems quite lacking. --129.138.220.140 (talk) 16:25, 7 October 2009 (UTC)

Yes, that seems reasonable. A formal proof wouldn't be too long so I'd say feel free to go ahead. Olaf Davis (talk) 17:17, 7 October 2009 (UTC)

A more formal proof of continuity on the irrational numbers, would be essential for the article to be considered quality. Possible sources: See the last few pages of Excellent proof for proofs of continuity on the irrationals and discontinuity on the rational numbers.

A second, Proof but with a Mistake is easier to follow but needs to be rearranged. It defines m, then later assumes a property about m. A better ordering starts,

Let epsilon > 0, Let m be an integer such that, 1/m < epsilon.

The second proof may have other mistakes. It could use a clearer transition from proof of continuity proof of discontinuity. Replacing variables in the 2nd half to prevent confusion with the first half, would also be useful. The next step would be to investigate copyright issues for these proofs. Or possibly rewrite them to avoid copyright issues. --Nickalh50 (talk) 19:41, 14 March 2011 (UTC)

Hi guys. I rewrote the proofs and put them in collapse boxes. I believe that the proofs are complete and precise now. Copyright problems shouldn't be an issue; I used only this article as a guide. (Any constructive criticism on my math writing would be greatly appreciated.)
Actually I started an hour or two ago by deleting the "Integrability" section, since it made the obviously false statement that the Thomae function is only discontinuous on the rationals. I'm wiser now. 04:18, 9 September 2017 (UTC) — Preceding unsigned comment added by Norbornene (talkcontribs)

## Naming convention?

In God Created the Integers, Stephen Hawking refers to what is defined here as the Dirichlet function as the "characteristic function of the rationals in the reals", and instead calls this function the "Dirichlet function". Wolfram MathWorld defines both of them under the same banner "Dirichlet function", where the function may take the two values (for rational and irrational) as additional parameters; it calls this function the "modified Dirichlet".

I remain uncertain that either "Thomae's function" or "popcorn function" is a proper canonical name if it is not the most common name in the literature. At the very least it should be mentioned that this function is quite often called "the Dirichlet function", regardless as to its attribution (which is correctly given to Thomae). TricksterWolf (talk) 00:42, 26 August 2011 (UTC)

## Amendments and tweaks

I tweaked the lede, added a few properties, and tweaked the proofs. I tried to keep the proofs as elementary epsilon/delta proofs, and also to be constructive and rigorous. Purgy (talk) 10:15, 19 September 2017 (UTC)