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Interval Analysis and Calculus for Interval-Valued Functions of a Single Variable. Part I: Partial Orders, gH-Derivative, Monotonicity

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Department of Economics, Society, Politics, University of Urbino Carlo Bo, Via A. Saffi 42, 61029 Urbino, Italy
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Department of Statistical Sciences “Paolo Fortunati”, University of Bologna, 40126 Bologna, Italy
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Author to whom correspondence should be addressed.
Axioms 2019, 8(4), 113; https://doi.org/10.3390/axioms8040113
Received: 19 July 2019 / Revised: 2 October 2019 / Accepted: 3 October 2019 / Published: 14 October 2019
We present new results in interval analysis (IA) and in the calculus for interval-valued functions of a single real variable. Starting with a recently proposed comparison index, we develop a new general setting for partial order in the (semi linear) space of compact real intervals and we apply corresponding concepts for the analysis and calculus of interval-valued functions. We adopt extensively the midpoint-radius representation of intervals in the real half-plane and show its usefulness in calculus. Concepts related to convergence and limits, continuity, gH-differentiability and monotonicity of interval-valued functions are introduced and analyzed in detail. Graphical examples and pictures accompany the presentation. A companion Part II of the paper will present additional properties (max and min points, convexity and periodicity). View Full-Text
Keywords: interval-valued functions; monotonic interval functions; comparison index; partial orders; lattice of real intervals; interval calculus interval-valued functions; monotonic interval functions; comparison index; partial orders; lattice of real intervals; interval calculus
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Stefanini, L.; Guerra, M.L.; Amicizia, B. Interval Analysis and Calculus for Interval-Valued Functions of a Single Variable. Part I: Partial Orders, gH-Derivative, Monotonicity. Axioms 2019, 8, 113.

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