"Continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function."
Continuity is the primary concept in topology, it is defined as a function between two topological spaces that preserves their underlying topological structure. A homeomorphism is a bijective continuous function, its inverse is also continuous. Two spaces are homeomorphic if they can be transformed into each other through a homeomorphism.
Topology: The study of properties of spaces that are preserved under continuous transformations.
Metric Spaces: A space in which a notion of distance is defined between every pair of points.
Continuity: A notion that captures the idea of "no sudden jumps" or "no holes" in a function when one moves from one point to another.
Topological Spaces: A generalization of a metric space that abstracts away the notion of distance and focuses on the properties that are preserved under continuous transformations.
Limits and Convergence: The idea that a sequence of points in a space can converge to a limit point, and the notion of a function being continuous at a point.
Open and Closed Sets: Sets that determine the topology of a space, and the relationships between them.
Connectedness and Path Connectivity: The idea of connectedness, in which a space cannot be split into disjoint pieces, and the stronger property of path connectivity, in which there exists a path between any two points in the space.
Homeomorphisms and Topological Equivalence: A map between two spaces that preserves the topological structure of the spaces, and the notion of two spaces being equivalent if there exists a homeomorphism between them.
Compactness: A property of a space that captures the idea of "finiteness" in some sense, and has important applications in many areas of mathematics, including analysis and geometry.
Separation Axioms: A set of axioms that define the "separation" properties of topological spaces, such as the Hausdorff and Urysohn axioms.
Embeddings and Immersions: A map between two spaces that preserves certain properties of the spaces, such as the topology or differentiability, and the stronger notion of an immersion, which is a map that preserves the differential structure of the spaces.
Manifolds: A special type of topological space that is locally Euclidean, and has important applications in many areas of mathematics and physics.
Pointwise continuity: A function f: X -> Y is pointwise continuous if and only if for every x in X and every ε > 0, there exists a δ > 0 such that d_Y(f(x),f(y)) < ε for all y in X with d_X(x,y) < δ.
Uniform continuity: A function f: X -> Y is uniformly continuous if and only if for every ε > 0, there exists a δ > 0 such that d_Y(f(x),f(y)) < ε for all x,y in X with d_X(x,y) < δ.
Lipschitz continuity: A function f: X -> Y is Lipschitz continuous if and only if there is a constant K > 0 such that d_Y(f(x),f(y)) ≤ Kd_X(x,y) for all x,y in X.
Hölder continuity: A function f: X -> Y is Hölder continuous if and only if there exists a constant K > 0 and α ∈ (0,1] such that d_Y(f(x),f(y)) ≤ Kd_X(x,y)^α for all x,y in X.
Continuous on a subset: A function f: S -> Y is continuous on a subset A of S if and only if for every ε > 0, there exists a δ > 0 such that d_Y(f(x),f(y)) < ε for all x,y in A with d_S(x,y) < δ.
Topological homeomorphism: A function f: X → Y is a homeomorphism if it is a one-to-one and onto continuous function with a continuous inverse.
Quasi-homeomorphism: A function f: X → Y is a quasi-homeomorphism if there exists a constant K > 0 such that for every ε > 0, there is a δ > 0 such that for all x,y in X with d_X(x,y) < δ, K^{-1}d_Y(f(x),f(y)) < d_X(x,y)^α < Kd_Y(f(x),f(y)).
Local homeomorphism: A function f: X → Y is a local homeomorphism if for every x in X, there exists a neighborhood U of x such that the restriction of f to U is a homeomorphism onto its image.
Brouwer homeomorphism: A function f: X → Y is a Brouwer homeomorphism if it is a continuous bijection from X onto Y and for every A ⊂ X, f(cl(A)) = cl(f(A)).
Conformal mapping: A function f: X → Y is a conformal mapping if it preserves angles. More precisely, if f is holomorphic and bijective, and if the derivative f' is everywhere nonzero, then f is conformal.
"Discontinuities are abrupt changes in value."
"A function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument."
"Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions."
"The epsilon–delta definition of a limit was introduced to formalize the definition of continuity."
"Continuity is one of the core concepts of calculus and mathematical analysis."
"The concept has been generalized to functions between metric spaces and between topological spaces."
"A stronger form of continuity is uniform continuity."
"In domain theory, a related concept of continuity is Scott continuity."
"An example, the function H(t) denoting the height of a growing flower at time t would be considered continuous."
"In contrast, the function M(t) denoting the amount of money in a bank account at time t would be considered discontinuous, since it 'jumps' at each point in time when money is deposited or withdrawn."