### Abstract

When investigating utilities, one of the most frequent assumptions is that of monotonicity, which on the real line implies differentiability almost everywhere (a.e.). In this note we examine this well-known property (due to Lebesgue) of real functions (utility functions). We prove that this result remains valid if a real function is monotonic except for a (countable) set of isolated points. Differentiability a.e. also holds if we augment the set of isolated points with their points of accumulation given that those are isolated from each other. If monotonicity fails on a set of points which is dense on a set with positive measure then the differentiability result is no longer necessarily valid.

Original language | English (US) |
---|---|

Pages (from-to) | 922-926 |

Number of pages | 5 |

Journal | International Journal of Mathematical Education in Science and Technology |

Volume | 24 |

Issue number | 6 |

DOIs | |

State | Published - 1993 |

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### ASJC Scopus subject areas

- Mathematics (miscellaneous)
- Applied Mathematics
- Education

### Cite this

*International Journal of Mathematical Education in Science and Technology*,

*24*(6), 922-926. https://doi.org/10.1080/0020739930240616

**A note on when monotonicity implies differentiability a.e.** / Dror, Moshe; Hartman, Bruce C.

Research output: Contribution to journal › Article

*International Journal of Mathematical Education in Science and Technology*, vol. 24, no. 6, pp. 922-926. https://doi.org/10.1080/0020739930240616

}

TY - JOUR

T1 - A note on when monotonicity implies differentiability a.e

AU - Dror, Moshe

AU - Hartman, Bruce C.

PY - 1993

Y1 - 1993

N2 - When investigating utilities, one of the most frequent assumptions is that of monotonicity, which on the real line implies differentiability almost everywhere (a.e.). In this note we examine this well-known property (due to Lebesgue) of real functions (utility functions). We prove that this result remains valid if a real function is monotonic except for a (countable) set of isolated points. Differentiability a.e. also holds if we augment the set of isolated points with their points of accumulation given that those are isolated from each other. If monotonicity fails on a set of points which is dense on a set with positive measure then the differentiability result is no longer necessarily valid.

AB - When investigating utilities, one of the most frequent assumptions is that of monotonicity, which on the real line implies differentiability almost everywhere (a.e.). In this note we examine this well-known property (due to Lebesgue) of real functions (utility functions). We prove that this result remains valid if a real function is monotonic except for a (countable) set of isolated points. Differentiability a.e. also holds if we augment the set of isolated points with their points of accumulation given that those are isolated from each other. If monotonicity fails on a set of points which is dense on a set with positive measure then the differentiability result is no longer necessarily valid.

UR - http://www.scopus.com/inward/record.url?scp=84946324848&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84946324848&partnerID=8YFLogxK

U2 - 10.1080/0020739930240616

DO - 10.1080/0020739930240616

M3 - Article

AN - SCOPUS:84946324848

VL - 24

SP - 922

EP - 926

JO - International Journal of Mathematical Education in Science and Technology

JF - International Journal of Mathematical Education in Science and Technology

SN - 0020-739X

IS - 6

ER -