### 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 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

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## Cite this

Preiss, D., & Saito, S. (2014). Knot points of typical continuous functions.

*Transactions of the American Mathematical Society*,*366*(2), 833-856. https://doi.org/10.1090/S0002-9947-2013-06100-4