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In geometry, a set of points is convex if it contains every line segment between two points in the set. Equivalently, a convex set or a convex region is a set that intersects every line in a line segment, single point, or the empty set. [1][2] For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example ...
In mathematics, real is used as an adjective, meaning that the underlying field is the field of the real numbers (or the real field). For example, real matrix, real polynomial and real Lie algebra. The word is also used as a noun, meaning a real number (as in "the set of all reals").
Interval (mathematics) The addition x + a on the number line. All numbers greater than x and less than x + a fall within that open interval. In mathematics, a real interval is the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the ...
Thomae's function. Function that is discontinuous at rationals and continuous at irrationals. Point plot on the interval (0,1). The topmost point in the middle shows f (1/2) = 1/2. Thomae's function is a real -valued function of a real variable that can be defined as: [1]: 531.
The set of natural numbers is a proper subset of the set of rational numbers; likewise, the set of points in a line segment is a proper subset of the set of points in a line. These are two examples in which both the subset and the whole set are infinite, and the subset has the same cardinality (the concept that corresponds to size, that is, the ...
Real analysis is an area of analysis that studies concepts such as sequences and their limits, continuity, differentiation, integration and sequences of functions. By definition, real analysis focuses on the real numbers, often including positive and negative infinity to form the extended real line.
As a concrete example of this, if U is defined as the set of rational numbers in the interval (,), then U is an open subset of the rational numbers, but not of the real numbers. This is because when the surrounding space is the rational numbers, for every point x in U , there exists a positive number a such that all rational points within ...
With these definitions in hand, Cantor's isomorphism theorem states that every two unbounded countable dense linear orders are order-isomorphic.[1] Within the rational numbers, certain subsets are also countable, unbounded, and dense. The rational numbers in the open unit interval are an example. Another example is the set of dyadic rational ...