Stetiges

Stetiges is a German term associated with the mathematical concept of continuity. In German mathematics, the core adjective is stetig, and the corresponding noun use is usually expressed as stetige Funktionen (continuous functions). The word itself conveys the idea that a quantity changes without jumps or abrupt breaks, a notion that translates into the standard mathematical idea of continuity.

In formal terms, continuity can be defined in several equivalent ways. In topology, a function f from a space X to a space Y is continuous if the preimage of every open set in Y is open in X. In metric or normed spaces this is often stated via the epsilon-delta condition: for every point x0 and every epsilon greater than zero, there exists a delta greater than zero such that whenever the distance between x and x0 is less than delta, the distance between f(x) and f(x0) is less than epsilon. This captures the intuitive idea that small changes in input lead to small changes in output.

A function may be continuous at a point or on a set. Common examples of continuous functions on the real numbers include the identity function f(x) = x and polynomial functions like f(x) = x^2. Discontinuous examples include functions with jump discontinuities such as the Heaviside step function at 0 or the Dirichlet function, which assigns different values on rationals and irrationals.

Key properties include that the composition of continuous functions is continuous, and that a function continuous on a compact set is bounded and attains extrema (the Extreme Value Theorem). Moreover, a sequence of functions that converges uniformly to a continuous limit yields a continuous limit function.

In summary, stetiges describes the property of continuity in German mathematical language, a foundational concept in analysis and topology.