Spojitá

Spojité is a term used in Czech and Slovak mathematics to describe the property of continuity. It is applied to functions, curves, and mappings to indicate that they have no abrupt breaks; small changes in input produce small changes in output.

In the real-valued case, a function f: R → R is spojité at a point x0 if lim_{x→x0} f(x) = f(x0). A function is spojité on a set if it is spojité at every point of the set. The epsilon–delta formulation states that for every ε > 0 there exists δ > 0 such that |x − x0| < δ implies |f(x) − f(x0)| < ε. In metric or topological spaces, a function is spojité at x0 when small changes in the input lead to small changes in the output, in a way that can be described with the standard continuity definition.

Equivalently, in the sequential form: for every sequence x_n → x0, we have f(x_n) → f(x0). This provides another way to characterize spojitost in familiar real analysis.

Examples include polynomials, sine, and the exponential function, which are spojité on R. Functions with a jump, such as the step function at its jump, are not spojité at the point of discontinuity. The Dirichlet function is not continuous at any point where it is defined nontrivially.

Key consequences include that continuous functions preserve limits, the image of a connected interval is connected, and a continuous function on a compact set is uniformly continuous. The intermediate value theorem applies to functions that are spojité on an interval. Relation to differentiability is that every differentiable function is spojité, but a continuous function need not be differentiable. In topology, spojitost (continuity) is defined via preimages of open sets being open.