Càdlàg Function¶
A time-indexed function whose value agrees with its right limit and whose pre-time left limit exists, allowing jumps with a defined before-and-after state.
Core Idea¶
A càdlàg function is a real-interval-indexed path that is right-continuous and has a left limit at every point where that side of the domain is available. The value at a jump is the post-jump value, while the left limit records the pre-jump state. A continuous path also qualifies; a jump is permitted, not required.[^ref-2965b47bc575]
Scope of Application¶
MIT's three-fair-coin-toss example gives a heads-count CDF with values \(1/8,1/2,7/8,1\) on successive intervals starting at 0, 1, 2 and 3; the jump at 1 is \(3/8\), and the value includes that atom. MIT's Poisson count \(N(t)\) likewise includes an arrival occurring at \(t\), so its unit-jump path is right-continuous with a pre-arrival left limit. Such paths form the raw members of Skorokhod space, though choosing a topology on that space is a separate step.[ref-dda64abf43f9][ref-122fcab6c408][^ref-2965b47bc575]
Clarity¶
For \(f:[0,T]\to M\), the tests are \(f(t)=\lim_{s\downarrow t}f(s)\) where a right approach exists, and existence of \(\lim_{s\uparrow t}f(s)\) where a left approach exists. A left-continuous step that retains the old value at its jump fails the first test. Càdlàg path regularity does not by itself make a stochastic process adapted to a filtration.[ref-2965b47bc575][ref-96b9a631a79f]
Manages Complexity¶
The two one-sided conditions let analysts use before-and-after states even when a path jumps, without assuming a finite number of events. For real-valued paths on a compact interval, there can be countably many jumps. Whether paths are close under a particular topology, or whether a process has a specified probability law, requires more information.[ref-0c03c137e1c7][ref-2965b47bc575]
Abstract Reasoning¶
State the real interval and metric state space. Check right-continuity and existence of left limits at every applicable point, respecting endpoints. Classify a jump by comparing \(f(t-)\) with \(f(t)\); do not make jump occurrence or adaptedness part of the definition. To reason about stochastic convergence, separately declare the path-space topology and filtration.[^ref-2965b47bc575]
Knowledge Transfer¶
The same before/after path convention works for probability thresholds and event times, although the meaning of a jump differs: atom mass in a CDF, event increment in a counting process. The live Function (Mapping) is the strict genus; Continuous Function is a jump-free special case and Cumulative Distribution Function a probability-constrained member, not parents.
[^ref-2965b47bc575]: Encyclopedia of Mathematics, “Skorokhod space”. [^ref-96b9a631a79f]: ETH Zürich, semi-martingale theory lecture notes. [^ref-dda64abf43f9]: MIT 14.30, Lecture Note 2, Examples 3.1, 4.1, 4.4 and §4.2. [^ref-122fcab6c408]: MIT 6.436J, Lecture 20, Poisson counting path. [^ref-0c03c137e1c7]: Research on discontinuities of the first kind and countable jumps.
Relationships to Other Abstractions¶
Current abstraction Càdlàg Function Domain-specific
Parents (1) — more general patterns this builds on
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Càdlàg Function is a kind of Function (Mapping) Prime
A càdlàg path is a single-valued function with right continuity and left limits.
Hierarchy path (1) — routes to 1 parentless root
- Càdlàg Function → Function (Mapping)
Neighborhood in Abstraction Space¶
Càdlàg Function sits in a sparse region of the domain-specific corpus (63rd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Dynamical Systems & Differential Structures (37 abstractions)
Nearest neighbors
- Matrix Difference Equation — 0.86
- Slow-Growing Hierarchy — 0.85
- Continuous-Time Random Walk — 0.85
- Residence Time (Statistics) — 0.84
- Differential Inclusion — 0.84
Computed from structural-signature embeddings · 2026-10-08