### Abstract

A real valued function f defined on a real interval I is called d-Lipschitz if it satisfies \ℓ(x) - ℓ(y)\ ≦ d(x, y) for x, y ∈ I. In this paper, we investigate when a function p : I → R can be decomposed in the form p = q + ℓ, where q is increasing and ℓ is d-Lipschitz. In the general case when d : I^{2} → R is an arbitrary semimetric, a function p : I → R can be written in the form p = q + ℓ if and only if (Equation Presented) is fulfilled for all real numbers t_{1} < s _{1}, . . . , t_{n} < s_{n} and u_{1} < ν_{1}, . . , ν_{m} < ν_{m}, in I satisfying the condition (Equation Presented) where 1_{[a,b]} denotes the characteristic function of the interval _{]a, b]}. In the particular case when d : I^{2} → R is a so-called concave semimetric, a function p : I → R is of the form p = q + ℓ if and only if (Equation Presented) holds for all x_{0} ≦ x_{1} < ⋯< x_{2n} ≦ x_{2n+1} in I.

Original language | English |
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Pages (from-to) | 1-18 |

Number of pages | 18 |

Journal | Acta Mathematica Hungarica |

Volume | 113 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - Oct 1 2006 |

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### Keywords

- Lipschitz function
- Lipschitz perturbation
- Monotonicity

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Acta Mathematica Hungarica*,

*113*(1-2), 1-18. https://doi.org/10.1007/s10474-006-0086-9