continuitii

Continuitii is the Romanian plural form of continuitate, corresponding to the mathematical concept of continuity. In mathematics, continuity describes a relationship in which small changes in input produce small changes in output.

In metric spaces, a function f: X → Y is continuous at a point a if for every ε > 0 there exists δ > 0 such that d_X(x, a) < δ implies d_Y(f(x), f(a)) < ε. Equivalently, lim_{x→a} f(x) = f(a).

Topology provides a more general definition: f is continuous if the preimage of every open set in Y is open in X. In metric spaces this agrees with the ε–δ definition; sequences offer another characterization: if x_n → a then f(x_n) → f(a).

Uniform continuity strengthens this by requiring δ to depend only on ε, not on the point a. The Heine–Cantor theorem states that every continuous function on a compact space is uniformly continuous.

Discontinuities of a real function at a point can be classified as removable, jump, infinite, or oscillatory. A function is continuous on an interval if and only if it has no such discontinuities there.

Common continuous functions include polynomials, exponentials, logarithms (where defined), and trigonometric functions. The sum, product, and composition of continuous functions are continuous; discontinuities can arise from domain restrictions or piecewise definitions.

Historically, the formal study of continuity emerged in the 19th century with Cauchy, Bolzano, and Weierstrass, culminating in the epsilon–delta framework and the development of topology as a setting for continuity.