The weierstrass function is one of famous examples of pathological function. The property of that function is continuous for all real numbers, but not differentiable everywhere.

Before discovering the pathological function, every continuous function was considered to be differentiable.

I do not understand the thought that The continuous functions are differentiable. So,I do not know why the weierstrass function is pathological. Also,I would like to gain views of the weierstrass function from present mathematics.

I wish you answer this question.


closed as unclear what you're asking by Mauro ALLEGRANZA, Michael Hoppe, Lord Shark the Unknown, Vladhagen, Parcly Taxel Oct 23 '18 at 10:35

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  • 3
    $\begingroup$ "Before discovering the pathological function, every continuous function was considered to be differentiable." - This is not true. In fact there are very simple examples of functions which are continuous but not differentiable at some points, e.g. $x \mapsto |x|$, which is not differentiable at $x=0$. The reason why the Weierstrass function is so interesting is that it is continuous, but not differentiable at any point of the domain. Also, it is quite unclear what you are actually asking. $\endgroup$ – MisterRiemann Oct 19 '18 at 9:22
  • 1
    $\begingroup$ See Weierstrass function : "The function has the property of being continuous everywhere but differentiable nowhere. It is named after its discoverer Karl Weierstrass. Historically, the Weierstrass function is important because it was the first published example (1872) to challenge the notion that every continuous function is differentiable except on a set of isolated points." $\endgroup$ – Mauro ALLEGRANZA Oct 19 '18 at 9:53
  • $\begingroup$ In my opinion the word "pathological" should be banned from mathematics. Many people use it for objects (e.g. functions) that do not behave in the way they expected, i,e. disprove conjectures about certain mathematical objects. And sometimes "pathological" behavior is the typical case, see e.g. en.wikipedia.org/wiki/Pathological_(mathematics). $\endgroup$ – Paul Frost Oct 19 '18 at 11:54

I think the view of the Weierstrass function nowadays is

  1. It's not pathological – it's typical – there is a well-defined sense in which most everywhere continuous functions are nowhere differentiable,

  2. It has the advantage over other functions with the above properties of being easy enough to describe that even an undergraduate can understand it, so it's handy when, for whatever reason, you need a function with those properties,

  3. Other than that, no one really cares about it very much. I mean, there's a million dollars on offer for figuring out what the Riemann zeta function does; there are no prizes for settling open problems about the Weierstrass function.

  • $\begingroup$ People still refer to continuous nowhere differentiable functions as pathologies. A recent book calls them monsters. $\endgroup$ – Andrés E. Caicedo Oct 19 '18 at 12:55
  • $\begingroup$ @AndrésE.Caicedo I am not sure whether the word "monster" is that negative as "pathology". For example, in group theory, the Monster Group is also known as the "Friendly Giant". Anyway, wordings are not really questions about mathematics. $\endgroup$ – Paul Frost Oct 19 '18 at 16:57
  • $\begingroup$ @Paul The authors of the book I linked to use the name for historical reasons, but the original use was certainly negative. Hermite said that they caused him "terror and horror''. Poincaré was who first called them monsters, and said that Weierstrass work was "an outrage against common sense''. $\endgroup$ – Andrés E. Caicedo Oct 19 '18 at 17:21
  • $\begingroup$ (@Paul Anyway, I disagree that questions about "wordings" are not about mathematics. Choosing appropriate terminology is actually important, and when it is done correctly, it helps guide our intuitions. For people within the profession, this is something worthy of consideration. I agree that a casual student may not see the relevance.) $\endgroup$ – Andrés E. Caicedo Oct 19 '18 at 17:25
  • $\begingroup$ @Andrés Thank you, this is a very interesting historical remark. And I agree that notation is worth considering. $\endgroup$ – Paul Frost Oct 19 '18 at 17:39

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