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Reading Halmos' I want to be a mathematician, he mentions a continuous function without a tangent. Naturally, I was curious to see how such a function could possibly exist, and I imagined it to be some obscure function where versions of $|x|$ are defined on infinitely small intervals. Much to my surprise, the series representation of the function was beautiful, especially for someone who just went through power series in Calculus.

This lead me to another question: was this function discovered or constructed? In other words, did Weierstrass discover the function, as Wikipedia slightly implies, or did Weierstrass construct the function, as in solving a problem of the likes "Construct a function that is everywhere continuous but nowhere differentiable."?

Andrew Thompson
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    It pretty much boils down to the question if math itself is discovered or created. – Leo Mar 02 '14 at 19:58
  • Well, by creating, I mean willingly attempting to create such a function. By discovering, I mean discovering the function as a, possibly surprising, result of research with another goal. – Andrew Thompson Mar 02 '14 at 20:04
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    @user51387 I don't think so. The way I read it, the question asks "Did Weierstrass stumble over this by accident, i.e. did he somehow derive that series and then discover that it's continouos but not differentiable, or was he specifically looking for such a thing and constructed that series for that purpose?". I've alway assumed the latter, but I don't have any sources I could cite... – fgp Mar 02 '14 at 20:05
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    he discovered he could create the function. – James S. Cook Mar 02 '14 at 20:14
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    Morris Kline writes: "As far back as 1861, he [Weiersrtass] had affirmed in his lectures that any attempt to prove that differentiability follows from continuity must fail. He then gave the classic example of a continuous nowhere differentiable function to the Berlin Academy on July 18, 1872." Of course, this does not answer your question; but at the time, the problem of finding such a pathological function was present and actively worked on by several mathematicians. I surmise Weierstrass actively sought to find such a function, rather than stumbling upon one. – David Mitra Mar 02 '14 at 20:17
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    Incidentally, Weierstrass was not the first to find a nowhere differentiable continuous function. Or at least to announce the result. Charles Cellerier gave an example in 1860. Bolzano had one in 1834, unpublished. – David Mitra Mar 02 '14 at 20:23
  • @David Mitra: I'm pretty sure that Bolzano did not prove (or even claim) that his function is nowhere differentiable in the sense that we mean today. Rather, I think he only considered the issue of not being differentiable in any interval (i.e. the non-differentiability set is dense in the reals), and the distinction between being densely non-differentiable and being everywhere non-differentiable wasn't on anyone's mind at the time. See this October 2000 sci.math post for more details and related issues. – Dave L. Renfro Mar 07 '14 at 20:52

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