Abstract
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).
Highlights
The motivations for this paper are two-fold: on one side we intend to offer an updated state of art about the concepts, problems and the techniques in interval analysis (IA), with a specific focus on its mathematical aspects recently addressed by research; secondly, we aim to contribute directly to the theoretical aspects of calculus in the setting of interval-valued functions of a single real variable
We will not explicitly address the problems in interval arithmetic, in relation to the algebraic aspects of how arithmetical operations can be defined for intervals and how to solve problems such as algebraic, differential, integral equations or others when intervals are involved
By varying the two parameters −∞ ≤ γ− ≤ 0, 0 ≤ γ+ ≤ +∞, we obtain a continuum of partial order relations for intervals and we have the following equivalences [59]: Proposition 6
Summary
The motivations for this paper are two-fold: on one side we intend to offer an updated state of art about the concepts, problems and the techniques in interval analysis (IA), with a specific focus on its mathematical aspects recently addressed by research; secondly, we aim to contribute directly to the theoretical aspects of calculus in the setting of interval-valued functions of a single real variable. A, ...Z, Ai , ..., Zj ∈ KC in upper-case characters; when we refer to an interval C ∈ KC , its elements are denoted as c− , c+ , cb, ce, with ce ≥ 0 , c− ≤ c+ and the interval by C = [c− , c+ ] in extreme-point representation or, equivalently, by C = (cb; ce), in midpoint notation. The introduction of two additions ⊕ M , ⊕ gH and two differences M , gH for intervals is not motivated here as an attempt to define a “true" arithmetic in KC ; for example, ⊕ M and ⊕ gH are both commutative with neutral element 0, but only ⊕ M is associative.
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