cadlagrightcontinuous

Cadlagrightcontinuous is a term used to describe a class of functions that are central in analysis and probability theory. It is short for the French phrase “continue à droite, limites à gauche,” commonly rendered in English as “right-continuous with left limits.” A function is càdlàg (or cadlag) on a real interval if it is right-continuous at every point and, at every point, has a finite left-hand limit. In more precise terms, for a function f defined on [0, ∞) or an interval, for each t in the domain the right-hand limit lim_{s↓t} f(s) exists and equals f(t), and the left-hand limit lim_{s↑t} f(s) exists (often denoted f(t−)).

The concept generalizes to functions taking values in metric spaces and is widely used to describe sample paths of stochastic processes. In many probabilistic constructions, processes are assumed to be càdlàg to accommodate jumps while preserving a form of continuity from the right. The combination of right-continuity and the existence of left limits ensures that jumps occur discretely and are well-defined as the process evolves.

Common examples include step functions that are constant between jump times, and the sample paths of a Poisson process, which exhibit jumps at random times but are right-continuous with left limits. Continuous processes, such as Brownian motion, are also càdlàg since they are continuous (and thus satisfy the right-continuity and left-limit requirements).

In analysis and topology, the space of càdlàg functions on an interval, often denoted D([0, T], R) or similar, is equipped with topologies (notably the Skorokhod J1 topology) that make it a natural setting for convergence of stochastic processes.