constant function distance
A constant function f is one whose value is the same at all points of its domain.
Analysis: A function f is continuous at a point x of its domain if, whenever we are given a number e greater than 0, we may find a d greater than 0 so that whenever y is within a d-neighborhood of x, then f(y) is within an e-neighborhood of f(x).
Topology: A transformation of one topological space into another is continuous if the inverse image of every open set is open, or equivalently if the inverse image of every closed set is closed.
Cf. uniformly continuous, equicontinuous, absolutely continuous.
Naively, a region of space is convex if the line segement joining any two points of the region lies wholly within it. Thus, a polygon is convex if every line segment joining any two points on its sides lies entirely within the polygon. (This is equivalent to the condition that all its interior angles be less than 180°.)
More generally, a region in a real vector space is convex if whenever two points x and y are in the region then so is any point tx + (1 - t)y, where t lies in the interval [0, 1]. See the immediately following entries for additional uses of the descriptor “convex.”
A function is convex if the chord joining any two points of its graph lies entirely above the graph.
Cf. concave function.
A set is countable if it is finite, or if it is infinite and bijective to the set of natural numbers (finite ordinals), i.e., if there exists a complete one-to-one mapping of the set in question onto the set N. Sets that are both countable and infinite are sometimes called denumerable. Georg Cantor proved that sets may be uncountably infinite, for example the set of real numbers.
See vector product.
A proof by the 17th century French mathematician Germinal Dandelin of the equivalence of the plane-geometry and conic-section definitions of the ellipse, parabola, and hyperbola. See the article for a full exposition.
A conic section in which the intersecting plane passes through the vertex.
Given a space X and a subset A of X, we say that A is dense in X if the intersection of every open set of X with A is non-empty. E.g., the rational numbers are dense in the real numbers.
A set is denumerable if it is infinite and countable.
For a function f of a single real variable x, the derivative is defined as the limit of the difference quotient
provided this limit exists. In practice, the derivative is interpreted as the instantaneous rate of change of the function at x. Graphically, the derivative returns the slope of the tangent at x.
Cf. differentiation rules.
Given a set X, the derived set of X is the set of accumulation points of X. The second derived set is the derived set of the derived set, and so on.
Cf. Cantor-Bendixson Theorem.
Geometry: A diameter of a circle (or sphere) is a line containing the center and with endpoints on the perimeter (resp. surface).
Analysis: Given a set X in a metric space, the diameter of X is the supremum of the distances between all pairs of points of X.
Graph Theory: The diameter of a given graph G is the maximum, over all pairs of vertices u, v of G that are in the same connected component of G, of the distance between u and v. In other words, it is the greatest distance between two vertices on the graph.
A function is differentiable at a point of its domain if its derivative exists at that point. A function is said to be (simply) differentiable if its derivative exists at all points of its domain.
A rule permitting easy differentiation of functions having certain forms. See the article for a complete description.
A set of points consisting of a circle together with its interior points. The set consisting only of the interior points of a circle is called an open disk.
The distance between two points in a space is given by the length of the geodesic joining those two points. In Euclidean space, the geodesic is given by a straight line, and the distance between two points is the length of this line. The distance between two points a and b on a real number line is the absolute value of their difference, i.e., d(a, b) = |a - b|. In two (or more) dimensions, the distance is given by the (generalized) Pythagorean theorem, i.e., in a Cartesian coordinate system of n dimensions, where a = (a1, ... ,an) and b = (b1, ... ,bn), the distance d(a, b) is given by
The concept of distance may be generalized to more abstract spaces – such a distance concept is referred to as a metric.
Graph Theory: The length of the shortest path between two vertices of a graph. If there is no path between two vertices, their distance is defined to be infinite. The distance between two vertices v and u is denoted by d(v, u). In a connected graph, distance is a metric.