Contínuos

Contínuos is a term in Portuguese mathematics used to describe concepts characterized by continuity, most commonly continuous functions (funções contí­nuas). A function f: X → Y is contí­nua if small changes in the input produce small changes in the output. In topology this is defined by the condition that the preimage of every open set in Y is open in X; in metric spaces this can be stated equivalently as: for every a in X and every ε > 0 there exists δ > 0 such that d(x, a) < δ implies d(f(x), f(a)) < ε. A sequence x_n converging to a in X implies that f(x_n) converges to f(a) in Y, which is another common characterization of continuity.

A function is contí­nua on a set if it is contí­nua at every point of the set. The class of contí­nua functions is closed under addition, multiplication, and composition; that is, the sum, product, and composition of contí­nuas are contí­nuas. Continuous functions preserve limits: if x_n → l, then f(x_n) → f(l). They also preserve certain geometric and topological properties, such as mapping connected sets to connected sets, and mapping compact sets to compact sets in appropriate spaces. In real analysis on intervals, continuous functions also satisfy the intermediate value theorem and, on compact domains, the extreme value theorem.

Examples include polynomials, exponential and logarithmic functions (where defined), and trigonometric functions, all of which are contí­nuas on their respective domains. Discontinuities arise at points where a function is not defined or has a jump, such as the step function. Understanding contí­nuos is fundamental in calculus, analysis, and topology, linking local behavior to global structure.