Deriválhatók

Deriválhatók is a Hungarian term used in mathematics to describe objects, usually functions, that possess a derivative at a given point or on a domain. In short, a function is deriválható at a point if its derivative exists there; if this holds for every point of an interval or set, the function is deriválható on that interval or set.

For a real-valued function f defined on an interval I, f is deriválható at a point x0 ∈ I if the limit lim_{h→0} (f(x0+h) − f(x0)) / h exists. The value of this limit is the derivative f′(x0). If f is deriválható for all x in I, it is deriválható on I, and f′ is defined on I. In several variables, a function f: U ⊆ R^n → R^m is differentiable at a point when there exists a linear map that best linearizes f near that point; this generalizes the one-variable derivative.

Key properties include that deriválhatók are continuous at the points of differentiability, and differentiability implies a local linear approximation. However, differentiability is strictly stronger than continuity: a continuous function need not be deriválható, as seen in the absolute value function at zero.

Common examples: polynomials, exponentials, logarithms, and trigonometric functions are deriválhatók on their domains; the function f(x) = |x| is deriválható for x ≠ 0 but not at x = 0.

In usage, a function or a class of functions may be described as deriválhatók to indicate the existence of derivatives at the relevant points or regions.