intervalul

Intervalul, in Romanian mathematical usage, refers to the basic notion of an interval on the real number line. An interval is a set of real numbers that lie between two endpoints, capturing all values between them. For real numbers a and b with a ≤ b, several standard forms are used: the closed interval [a,b], which includes both endpoints; the open interval (a,b), which excludes them; and the half-open intervals [a,b) and (a,b]. When endpoints are not fixed, unbounded intervals such as (a, ∞), (−∞, b], and (−∞, ∞) are also common. A degenerate interval [a,a] contains the single point a.

Key properties: Intervals are connected and convex subsets of the real line. The length of a finite interval [a,b] is b − a. The boundary of [a,b] consists of the endpoints a and b. The intersection of any two intervals is an interval (possibly empty), while the union of two intervals is an interval only if the intervals overlap or touch.

Notation and uses: Interval notation describes solution sets of inequalities, domains of functions, and definite integrals, among other concepts. Intervals provide a convenient way to specify ranges and to discuss continuity, monotonicity, and convergence on a compact or extended real line.

Context and variants: While intervalul denotes the real-analytic notion in Romanian texts, the broader idea appears in many fields, including time intervals (durations) and musical intervals, each with its own specialized interpretation. In mathematics, the interval remains a fundamental, widely used construct.