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I’m studying functional series and right now we’re dealing with Weierstrass function. I was wondering if it’s the only known function with this property?

Calum Gilhooley
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Alicia Mascareñas
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    Actually, most continous functions have this property. For starters, take any $C^1$ function and add a small multiple of the Weierstrass function. That shows that such functions are at least dense in $C^0$. The Wiener measure is a probability measure in the space of continuous function (https://en.wikipedia.org/wiki/Classical_Wiener_space). With respect to this measure, almost all continuous functions are nondifferentiable at every point. – Henrik Schumacher Apr 02 '20 at 10:49
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    The Wiener measure is what drives Brownian motion. So almost every path of Brownian motion is nowhere differentiable (but it is always continuous). – Henrik Schumacher Apr 02 '20 at 10:57

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Further to Henrik Schumacher's comment, here is a list of textbooks that apply the Baire Category Theorem to prove that "most" continuous functions on a compact interval of $\mathbb{R}$ are nowhere differentiable. (I've made this post Community Wiki, in case anyone wants to add more references.)

  • Stephen Abbott, Understanding Analysis (2nd ed. 2015), Theorem 8.2.12.

  • G. Bachman & L. Narici, Functional Analysis (1966, repr. Dover 2000), section 6.3.

  • Douglas S. Bridges, Foundations of Real and Abstract Analysis (1998), section 6.3. [sic]

  • N. L. Carothers, Real Analysis (2000), pp.184-187.

  • John DePree & Charles Swartz, Introduction to Real Analysis (1988), pp.293-295.

  • J. Dieudonné, Treatise on Analysis, vol. II (1969, 1970), section 12.16, problem 17.

  • James Dugundji, Topology (1966), p.300f.

  • Steven A. Gaal, Point Set Topology (1964, 1966, repr. Dover 2009), p.288f.

  • T. W. Gamelin & R. E. Greene, Introduction to Topology (2nd ed. Dover 1999), Exercise 1.6.11.

  • D. J. H. Garling, A Course in Mathematical Analysis, vol. II (2013), Exercise 14.7.3.

  • Marek Jarnicki and Peter Pflug, Continuous Nowhere Differentiable Functions (2015), chapter 7.

  • James R. Munkres, Topology (2nd ed. 2000), section 49.

  • John C. Oxtoby, Measure and Category (2nd ed. 1980), chapter 11.

  • Charles C. Pugh, Real Mathematical Analysis (1st ed. 2002, 2003), section 4.7.

  • Stephen Willard, General Topology (1970, repr. Dover 2004), Theorem 25.5.

Dave L. Renfro
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Calum Gilhooley
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    @nomen: I've added the Oxtoby reference and Jarnicki/Pflug's book, the latter of which probably has the most complete (pun intended) treatment in a published book. Given my interests, I'll refrain from trying to add any more, since there are simply too many to know where to begin or end. For what it's worth, however, this rabbit hole goes much, much deeper than just "nowhere differentiable" --- see my answer to Generic Elements of a Set. – Dave L. Renfro Apr 02 '20 at 19:08