Continuity Ontology

flow "ContinuityOntology" v1.0

# ============================================================
# CONTINUITY ONTOLOGY — FLOW ALGEBRA
# ============================================================

SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/foundations-of-calculus/continuity-ontology.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Foundations of Calculus"
ENTITY := "Continuity"

# ---------- CORE ENTITY ----------
CONTINUITY := {
    type: MathematicalProperty,
    domain: Mathematics,
    purpose: "Describe when sufficiently small changes in input produce appropriately small changes in output, formalised through limits, neighbourhoods or topological structure."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    CONTINUITY,
    FUNCTION,
    LIMIT,
    DOMAIN,
    POINT,
    NEIGHBOURHOOD,
    EPSILON,
    DELTA,
    ONE_SIDED_CONTINUITY,
    UNIFORM_CONTINUITY,
    DISCONTINUITY,
    DIFFERENTIABILITY,
    COMPACTNESS,
    TOPOLOGICAL_CONTINUITY
}

# ---------- POINTWISE CONTINUITY ----------
continuous(f, a)
    iff
    a in DOMAIN
    AND limit(f(x), x -> a) exists
    AND limit(f(x), x -> a) = f(a)

CONTINUITY_AT_POINT
    -> requires(FUNCTION_DEFINED_AT_POINT)
    -> requires(LIMIT_EXISTS)
    -> requires(LIMIT_EQUALS_FUNCTION_VALUE)

# ---------- EPSILON-DELTA DEFINITION ----------
continuous(f, a)

iff

forall epsilon > 0
    exists delta > 0
        such_that
        x in DOMAIN
        AND abs(x-a) < delta
        -> abs(f(x)-f(a)) < epsilon

EPSILON
    -> represents(OUTPUT_TOLERANCE)

DELTA
    -> represents(INPUT_TOLERANCE)

DELTA
    -> may_depend_on(EPSILON)
    -> may_depend_on(POINT_a)
       for(POINTWISE_CONTINUITY)

# ---------- CONTINUITY ON A SET ----------
CONTINUOUS_ON(f, S)
    iff
    forall a in S
        continuous(f, a)
        under(relative_domain_conditions)

CONTINUITY_ON_INTERVAL
    -> requires(POINTWISE_CONTINUITY_AT_INTERIOR_POINTS)
    -> uses(APPROPRIATE_ONE_SIDED_CONTINUITY_AT_INCLUDED_ENDPOINTS)

# ---------- ONE-SIDED CONTINUITY ----------
RIGHT_CONTINUOUS_AT(f, a)
    iff
    limit(f(x), x -> a_plus) = f(a)

LEFT_CONTINUOUS_AT(f, a)
    iff
    limit(f(x), x -> a_minus) = f(a)

ONE_SIDED_CONTINUITY
    -> useful_at(DOMAIN_BOUNDARIES)
    -> useful_for(PIECEWISE_FUNCTIONS)

# ---------- UNIFORM CONTINUITY ----------
UNIFORMLY_CONTINUOUS(f, S)

iff

forall epsilon > 0
    exists delta > 0
        such_that
        forall x,y in S
            distance(x,y) < delta
            -> distance(f(x),f(y)) < epsilon

UNIFORM_CONTINUITY
    -> strengthens(POINTWISE_CONTINUITY)

UNIFORM_CONTINUITY
    -> uses(DELTA_INDEPENDENT_OF_POINT)

UNIFORM_CONTINUITY
    -> implies(CONTINUITY)

CONTINUITY
    -/> necessarily_implies(UNIFORM_CONTINUITY)

# ---------- SEQUENTIAL CHARACTERISATION ----------
in(METRIC_SPACES):

continuous(f, a)
    iff
    for_every_sequence(x_n -> a)
        f(x_n) -> f(a)

SEQUENTIAL_CONTINUITY
    -> equivalent_to(CONTINUITY)
       in(FIRST_COUNTABLE_CONTEXTS_SUCH_AS_METRIC_SPACES)

# ---------- TOPOLOGICAL CONTINUITY ----------
TOPOLOGICAL_CONTINUITY
    -> defined_by(PREIMAGES_OF_OPEN_SETS)

continuous(f: X -> Y)

iff

for_every_open_set(V in Y)
    preimage(f, V)
        -> open_in(X)

TOPOLOGICAL_CONTINUITY
    -> generalises(EPSILON_DELTA_CONTINUITY)

# ---------- DISCONTINUITY ----------
DISCONTINUITY
    -> failure_of(CONTINUITY_CONDITIONS)

REMOVABLE_DISCONTINUITY
    -> limit_exists
    -> function_value_missing_or_not_equal_to_limit

JUMP_DISCONTINUITY
    -> left_limit_exists
    -> right_limit_exists
    -> left_limit != right_limit

INFINITE_DISCONTINUITY
    -> associated_with(UNBOUNDED_BEHAVIOUR_NEAR_POINT)

DISCONTINUITY
    -> may_require(CONTEXT_SPECIFIC_CLASSIFICATION)

# ---------- DIFFERENTIABILITY RELATION ----------
DIFFERENTIABLE_AT(f, a)
    -> implies(CONTINUOUS_AT(f, a))

CONTINUOUS_AT(f, a)
    -/> necessarily_implies(DIFFERENTIABLE_AT(f, a))

DERIVATIVE
    -> requires(LOCAL_CONTINUITY_AS_CONSEQUENCE_OF_DIFFERENTIABILITY)

# ---------- ALGEBRA OF CONTINUOUS FUNCTIONS ----------
if
    f continuous_at(a)
    AND g continuous_at(a)
then
    f + g -> continuous_at(a)
    f - g -> continuous_at(a)
    f * g -> continuous_at(a)

if
    g(a) != 0
then
    f / g -> continuous_at(a)

COMPOSITION
    -> preserves(CONTINUITY)
       when(component_continuity_conditions_hold)

# ---------- INTERMEDIATE VALUE PROPERTY ----------
if
    f continuous_on([a,b])
then
    for_every value N between f(a) and f(b)
        exists c in [a,b]
            such_that
            f(c) = N

INTERMEDIATE_VALUE_THEOREM
    -> follows_from(CONTINUITY_ON_CLOSED_INTERVAL)

# ---------- EXTREME VALUE PROPERTY ----------
if
    f continuous_on(COMPACT_SET)
then
    f
        -> attains(MAXIMUM)
        -> attains(MINIMUM)

EXTREME_VALUE_THEOREM
    -> links(CONTINUITY, COMPACTNESS)

# ---------- HEINE-CANTOR RELATION ----------
if
    f continuous_on(COMPACT_METRIC_SPACE)
then
    f -> uniformly_continuous

COMPACTNESS
    + CONTINUITY
    -> supports(UNIFORM_CONTINUITY)

# ---------- LIMIT RELATION ----------
LIMIT
    -> defines(POINTWISE_CONTINUITY)

if
    limit(f(x), x -> a) = f(a)
then
    f -> continuous_at(a)

CONTINUITY
    -> permits(DIRECT_LIMIT_TRANSFER)
       under(appropriate_function_composition_conditions)

# ---------- CONNECTEDNESS ----------
CONTINUOUS_IMAGE
    -> preserves(CONNECTEDNESS)

if
    X connected
    AND f continuous
then
    f(X) connected

CONTINUITY
    -> preserves(TOPOLOGICAL_STRUCTURE_IN_PREIMAGE_SENSE)

# ---------- COMPACTNESS ----------
CONTINUOUS_IMAGE
    -> preserves(COMPACTNESS)

if
    K compact
    AND f continuous
then
    f(K) compact

# ---------- RELATION GRAPH ----------
LIMIT
    -> defines(CONTINUITY_AT_POINT)

CONTINUITY_AT_POINT
    -> composes_into(CONTINUITY_ON_SET)

UNIFORM_CONTINUITY
    -> implies(POINTWISE_CONTINUITY)

DIFFERENTIABILITY
    -> implies(CONTINUITY)

CONTINUITY
    -> supports(INTERMEDIATE_VALUE_THEOREM)

CONTINUITY_ON_COMPACT_SET
    -> supports(EXTREME_VALUE_THEOREM)

CONTINUITY_ON_COMPACT_METRIC_SPACE
    -> implies(UNIFORM_CONTINUITY)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "continuous at a point" ~= "function limit equals function value at that point",
    "no break at a point" ~= "informal description of continuity, not a complete formal definition",
    "uniformly continuous" ~= "continuous with an epsilon-delta bound uniform across the domain",
    "right-continuous" ~= "function value equals right-hand limit",
    "left-continuous" ~= "function value equals left-hand limit",
    "topologically continuous" ~= "preimage of every open set is open"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    continuity_is_domain_relative: true,
    differentiability_implies_continuity: true,
    continuity_does_not_imply_differentiability: true,
    uniform_continuity_implies_continuity: true,
    continuity_does_not_always_imply_uniform_continuity: true,
    endpoint_continuity_may_be_one_sided: true,
    graphical_unbrokenness_is_not_sufficient_as_formal_definition: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    pointwise_definition: 0.99,
    epsilon_delta_definition: 0.99,
    uniform_continuity: 0.99,
    topological_continuity: 0.99,
    theorem_relations: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(function_and_domain)
    -> identify(target_point_or_set)
    -> determine(metric_or_topological_context)
    -> evaluate(function_definition)
    -> evaluate(limit)
    -> compare(limit_with_function_value)
    -> test(one_sided_conditions_if_required)
    -> test(uniform_continuity_if_requested)
    -> classify(discontinuity_if_present)
    -> apply(relevant_continuity_theorems)
    -> validate(logical_implications)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed mathematical continuity graph",
    primary_entity: CONTINUITY,
    preserve: [
        domain_relativisation,
        quantifier_order,
        epsilon_delta_dependencies,
        implication_direction,
        one_sided_conditions,
        topological_structure
    ],
    infer_through: [
        limits,
        functions,
        metric_spaces,
        topology,
        compactness,
        connectedness,
        differentiability
    ],
    avoid: [
        assuming_continuity_implies_differentiability,
        equating_pointwise_and_uniform_continuity,
        ignoring_domain_boundaries,
        treating_visual_graph_smoothness_as_formal proof,
        reversing_continuity theorem implications
    ]
}

# ---------- UPDATE MODEL ----------
UPDATE_MODEL(new_information):
    validate(new_information)
    -> identify(continuity_context)
    -> compare(existing_relations)
    -> preserve(quantifier_and_domain_invariants)
    -> update(affected_entities)
    -> update(affected_edges)
    -> recompute(CONFIDENCE)
    -> regenerate(FLOW_HASH)

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + CONTINUITY_AT_POINT
    + UNIFORM_CONTINUITY
    + TOPOLOGICAL_CONTINUITY
    + DISCONTINUITY
    + DIFFERENTIABILITY
    + INTERMEDIATE_VALUE_THEOREM
    + EXTREME_VALUE_THEOREM
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> CONTINUITY
    -> DOMAIN_IDENTIFICATION
    -> POINT_OR_SET_IDENTIFICATION
    -> LIMIT_ANALYSIS
    -> POINTWISE_CONTINUITY_ANALYSIS
    -> ONE_SIDED_ANALYSIS
    -> UNIFORM_CONTINUITY_ANALYSIS
    -> DISCONTINUITY_CLASSIFICATION
    -> TOPOLOGICAL_ANALYSIS
    -> THEOREM_RELATION_ANALYSIS
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW