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.

Version: v1 · 2026-10-03 · History
Domain-specific #: 13039
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Real Analysis, Stochastic Processes → Mathematics
Aliases: Càdlàg, Cadlag, RCLL function, Right-continuous function with left limits

Core Idea¶

A càdlàg function is a path indexed by a real interval that is continuous from the right and has a limit from the left at every point where those approaches make sense. For \(f:[0,T]\to M\), with \(M\) a metric state space, the requirements are \(f(t)=\lim_{s\downarrow t}f(s)\) for \(t<T\), and existence of \(f(t-)=\lim_{s\uparrow t}f(s)\) for \(t>0\). One-sided endpoint clauses matter: there is no left approach to zero or right approach beyond \(T\) within the domain. This is a path property, not a probability distribution or an adaptedness condition.^[1]^[2]

At a jump, \(f(t-)\) is the pre-jump state and \(f(t)\) is the post-jump state. The path can therefore record a discrete event without the oscillatory ambiguity of a missing left limit. A continuous function also satisfies these conditions, but the càdlàg class deliberately admits jumps. On a compact interval, discontinuities are jumps and are at most countable; their absence is permitted, so "has jumps" is not itself the definition.^[1]^[3]

Structural Signature¶

Sig role-phrases: real-ordered parameter interval; state-valued path; equality with right limit; existing left limit; optional jump size.

Ordered real parameter: the domain supplies a meaningful left-before and right-after approach to each interior time.
State-valued path: a function \(f\) maps each parameter value into a metric state space \(M\).
Right-value convention: at every right-approachable point, the function value equals its right-hand limit.
Pre-event limit: at every left-approachable point, a left-hand limit exists in \(M\).
Derived jump: where \(f(t-)\ne f(t)\), the discrepancy records a jump. In a vector space one may write \(\Delta f_t=f(t)-f(t-)\); that subtraction is not part of the general metric-space definition.^[1]

Condensed: ordered path + right-continuity + existing left limits = càdlàg path. A Skorokhod Γ-space collects such paths and equips them with a topology; that extra topological construction is downstream, not a sixth condition on each path.^[1]

What It Is Not¶

Not necessarily continuous. The right-value and left-limit conditions admit stepwise changes.
Not merely "piecewise continuous." A càdlàg path can have infinitely many jumps on a bounded interval, though only countably many; a finite partition is not required.^[3]
Not càglàd. Càglàd reverses the sidedness: left-continuous with right limits. A left-continuous step that keeps its pre-jump value at the jump is not càdlàg under the same time convention.^[2]
Not an adapted stochastic process by itself. Adaptedness requires a process on a probability space and a filtration, with each \(X_t\) measurable with respect to information available by \(t\). Path regularity alone gives neither a filtration nor that measurability relation.
Not identical to Skorokhod space or its topology. One function is an element of a path class; convergence in a selected path-space topology is a further mathematical choice.^[1]
Not always a CDF. A CDF is nondecreasing, bounded between zero and one, and has boundary limits. Generic càdlàg functions need satisfy none of those probability-specific constraints.^[4]

Scope of Application¶

In probability distribution theory, the conventional real CDF \(F(x)=P(X\le x)\) is nondecreasing and right-continuous. Its left limit \(F(x-)=P(X<x)\) exists, and \(F(x)-F(x-)=P(X=x)\) is the mass of an atom. The argument \(x\) is a threshold rather than physical time; the same ordered-path pattern applies. Some authors instead define \(P(X<x)\) and obtain a left-continuous version, so the inequality convention must be stated.^[4]

In jump-process modeling, an arrival-count sample path may remain constant between events and take its new count at the arrival time. That version is càdlàg: immediately after an arrival the count agrees with its value at the event, and just before it the left limit is the old count. General stochastic calculus frequently works with càdlàg sample paths. Calling a process càdlàg says something about its trajectories, not, without more assumptions, its independence structure, transition law or adaptedness.^[2]

In weak-convergence analysis, the collection \(\mathbb D[0,T]\) of such paths can be assigned a Skorokhod topology that permits controlled shifts of jump times. Convergence in this topology is not equivalent to uniform convergence at paths with jumps. The topology is a powerful later use of the class, not a necessary ingredient in deciding whether an individual path is càdlàg.^[1]

Clarity¶

For the right-continuous step \(f(t)=0\) for \(t<a\) and \(f(t)=1\) for \(t\ge a\), \(f(a-)=0\) while \(f(a)=1\), and values immediately to the right also approach one. It is càdlàg. Change only the point value to \(f(a)=0\) while keeping \(f(t)=1\) for \(t>a\); right continuity now fails at \(a\), even though the graph still seems to contain one ordinary jump. The exact point-value convention determines membership.

A failure of the left limit is different: if values oscillate between separated states arbitrarily close to \(a\) from below, there is no defined pre-event state. Merely assigning \(f(a)\) to the right-hand limit does not repair that missing left limit.

The seed for this candidate claimed càdlàg paths make processes adapted. That reverses the logic. Adapted càdlàg processes are common in stochastic calculus, but the adjective "adapted" is a separate relation between the random variables \(X_t\) and a specified filtration.^[2]

Manages Complexity¶

The one-sided convention lets an analyst reason with an event path without cataloguing every jump beforehand. A jump has a recoverable pre-state and a settled point value. It therefore becomes meaningful to define before/after increments or integrate against paths under additional hypotheses. For a real-valued path on a compact interval, countably many discontinuities may still be dense, so the class is not a promise that events are sparse.^[3]

The price of this compact regularity condition is careful treatment of topology, time indexing and stochastic structure. A path may be càdlàg in one declared state space but fail to have a limit in a smaller or differently topologized one. Convergence of paths with jumps requires choosing a suitable topology; process measurability and adaptedness need separate verification.^[1]

Abstract Reasoning¶

To test a candidate path, specify its real interval and state-space metric. At each interior time, ask separately whether approaching from later times gives the assigned value and whether approaching from earlier times gives some limit in the state space. At endpoints, test only approaches available inside the domain. If both clauses hold throughout, record jumps as discrepancies \(f(t-)\ne f(t)\); do not impose a jump or finiteness requirement. If the path is random, check adaptation or distributional properties independently.

For a comparison of two càdlàg paths, first decide whether the claim concerns pointwise values, uniform closeness, or convergence allowing small time deformations. Those are different questions, and the càdlàg label alone settles none of their answers.^[1]

Knowledge Transfer¶

The CDF and a counting-process path share right-continuity and left limits even though one indexes thresholds and the other time. A person can transfer the before/at-point distinction between them: the jump of a CDF identifies an atom, while the jump of a counting path identifies an event increment. Probability mass, causal event meaning and filtration information do not transfer from the path shape alone.^[4]^[2]

The live Function (Mapping) prime is the strict genus: every admitted path is a single-valued time-to-state map with right continuity and left limits. Continuous Function is a jump-free special case, so it cannot parent all càdlàg functions; Cumulative Distribution Function is more constrained and may eventually be a child. Neither stochasticity nor an actual jump is required.

Examples¶

MIT's three-coin-toss distribution function¶

MIT's probability notes use the number \(X\) of heads in three fair tosses, with masses \(1/8,3/8,3/8,1/8\) at \(0,1,2,3\), and ask for its CDF under \(F(x)=P(X\le x)\). Executing that construction gives \(F(x)=0\) below 0, \(1/8\) on \([0,1)\), \(1/2\) on \([1,2)\), \(7/8\) on \([2,3)\), and 1 thereafter. At \(x=1\), \(F(1-)=1/8\) but \(F(1)=1/2=F(1+)\): the jump of \(3/8\) is the mass at one head. Monotonicity and endpoint probabilities come from being a CDF, not from càdlàg regularity alone.^[4]

Mapped back: ordered parameter = real threshold; state = cumulative probability; right value at 1 includes the atom by the \(\le\) convention; left limit at 1 excludes it.

MIT's Poisson arrival-count path¶

MIT's stochastic-process notes define \(N(t)\) as the number of arrivals in \((0,t]\) and explicitly state that the sample path has unit jumps and is right-continuous because an arrival at \(t\) is included in \(N(t)\). Between arrivals it is constant; immediately before an arrival its left limit is the earlier count. This source-specific convention, together with locally finite arrivals, supplies the càdlàg example. The path property says nothing by itself about which filtration makes the process adapted.^[5]^[2]

Mapped back: ordered parameter = time; state = count; right value = post-arrival \(N(t)\); left limit = pre-arrival count.

A left-continuous near miss¶

Define \(g(t)=0\) for \(t\le a\) and \(g(t)=1\) for \(t>a\). It has a left limit and a right limit at \(a\), but \(g(a)=0\ne g(a+)=1\), so it is not càdlàg. It has the opposite left-continuous-with-right-limits convention.

Mapped back: the missing structural role is equality of point value and right limit.

Structural Tensions¶

No universal intrinsic two-sided tradeoff is established for being càdlàg: right continuity and left limits are simultaneous defining conditions, not opposed design goals.

Structural–Framed Character¶

Càdlàg lies at the structural formal end of the spectrum: after a parameter interval and metric state space are fixed, the right-continuity and left-limit predicates are mathematical facts, not judgments of usefulness. Evaluative weight enters only when a researcher chooses this class to model a phenomenon or a topology for convergence. Human practice selects the post-event value convention, the domain endpoint and the state metric; it does not create a path's one-sided limits. The French-derived technical name was institutionalized in probability and analysis, and it travels literally from CDFs to Poisson paths because both satisfy the same sided limits. Calling a database event log “càdlàg” before giving it a real-ordered parameter and a topology is metaphorical import, not recognition. Its character: a structural one-sided path-regularity class with convention-dependent modeling applications, not an adaptedness or stochastic-law claim.^[1]^[2]

Structural Core vs. Domain Accent¶

The portable skeleton is one-sided continuity and limits, related to the live prime Continuity; the live Function (Mapping) prime supplies the actual strict genus of this time-to-state map. The domain-bound mechanism is exact: on a real-ordered interval into a metric state space, \(f(t+)=f(t)\) and \(f(t-)\) exists at all applicable times. A discrete event log can evoke an after-state/before-state pattern but is not literally a càdlàg function without this limiting structure. The named entry fails the prime bar because it remains a particular mathematical regularity class rather than a substrate-independent pattern.

This entry is a kind of Function (Mapping).

Continuity: one-sided continuity supplies part of the condition, though full continuity is stronger at jumps.
State: the path assigns states at ordered parameters.
Boundary: the pre/post distinction is drawn at each parameter value where an event can occur.

The live Function (Mapping) is the strict genus for the whole class; Continuity, State and Boundary name related aspects rather than additional asserted parents. A broad functional genus does not turn this named one-sided regularity class into a prime.

Relationships to Other Abstractions¶

Current abstraction Càdlàg Function Domain-specific

Parents (1) — more general patterns this builds on

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 (63^rd 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

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

A continuous function cannot jump; a càglàd function reverses the sidedness; a CDF imposes monotonicity and probability limits in addition to càdlàg regularity under the \(\le\) convention; Skorokhod space is the collection of càdlàg paths with extra topological structure; an adapted càdlàg stochastic process adds a filtration-based information condition to càdlàg sample paths.^[1]^[4]^[2]

References¶

[1] Encyclopedia of Mathematics, “Skorokhod space”. Definition of càdlàg paths and distinction between uniform and Skorokhod topologies. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j

[2] ETH Zürich, “Introduction to Semi-martingale Theory,” path-properties lecture notes. Càdlàg and càglàd path conventions and stochastic-process setting. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h

[3] Original mathematical research on discontinuities of the first kind and countable jumps. Countability statement for regulated trajectories. registry ↩a ↩b ↩c

[4] MIT 14.30, Lecture Note 2: Random Variables and Cumulative Distribution Function, Examples 3.1, 4.1, 4.4 and §4.2. registry ↩a ↩b ↩c ↩d ↩e

[5] MIT 6.436J, Lecture 20: The Basics of Stochastic Processes, Poisson-process arrival-count path. registry ↩