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?
2026-04-03 01:23:12.1775179392
Is there any other function similiar to Weierstrass function, that is continuous at every point but non-differentiable at any point on a compact?
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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.