### Abstract

It is well known that most continuous functions are nowhere differentiable. Furthermore, in terms of Dini derivatives, most continuous functions are nondifferentiable in the strongest possible sense except in a small set of points. In this paper, we completely characterise families S of sets of points for which most continuous functions have the property that such small set of points belongs to S. The proof uses a topological zero-one law and the Banach-Mazur game.

Original language | English |
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Pages (from-to) | 833-856 |

Number of pages | 24 |

Journal | Transactions of the American Mathematical Society |

Volume | 366 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jan 1 2014 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

*Transactions of the American Mathematical Society*,

*366*(2), 833-856. https://doi.org/10.1090/S0002-9947-2013-06100-4

**Knot points of typical continuous functions.** / Preiss, David; Saito, Shingo.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 366, no. 2, pp. 833-856. https://doi.org/10.1090/S0002-9947-2013-06100-4

}

TY - JOUR

T1 - Knot points of typical continuous functions

AU - Preiss, David

AU - Saito, Shingo

PY - 2014/1/1

Y1 - 2014/1/1

N2 - It is well known that most continuous functions are nowhere differentiable. Furthermore, in terms of Dini derivatives, most continuous functions are nondifferentiable in the strongest possible sense except in a small set of points. In this paper, we completely characterise families S of sets of points for which most continuous functions have the property that such small set of points belongs to S. The proof uses a topological zero-one law and the Banach-Mazur game.

AB - It is well known that most continuous functions are nowhere differentiable. Furthermore, in terms of Dini derivatives, most continuous functions are nondifferentiable in the strongest possible sense except in a small set of points. In this paper, we completely characterise families S of sets of points for which most continuous functions have the property that such small set of points belongs to S. The proof uses a topological zero-one law and the Banach-Mazur game.

UR - http://www.scopus.com/inward/record.url?scp=84888093362&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84888093362&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-2013-06100-4

DO - 10.1090/S0002-9947-2013-06100-4

M3 - Article

VL - 366

SP - 833

EP - 856

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 2

ER -