Use the ordinary real open interval (-3,3) with left<right and constant map value=1 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).
G001 — Use the ordinary real open interval (-3,3) with left<right and constant map value=1 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).
Definition and scope
Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.
Retain the historical interface marker and list ordered source fields without JavaScript.MVT fields come from the original mode,without padding or answer-shape inference.Tags and fixed No markers retain their claim scope.
ORIGINAL WHOLE real interval(-3,3);constant real image value=1
After
for integers j>=0,relative opens U_j=(L+(R-L)/(j+2),R) cover the interval;ANY finite subfamily is contained in U_max_j and misses a nonempty interval near L,so domain NOT compact
Conditions: none
For every x>L,the Archimedean property gives j whose threshold is below x,so the whole family covers.Any finite family has a largest j and misses points between L and its positive threshold.This is an actual cover argument,not finite sampling or a nonclosed slogan.
preimage of ANY target open is either the whole domain or empty,so continuous;image singleton has a one-member subcover from any cover;reverse implication claim=No
Conditions: none
Both possible preimages of a constant map are open.A cover of a singleton has a member containing its point.Thus a compact image does not force a compact domain,nor general compact preimages or closed images without further hypotheses.
S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
Result
(-3, 3, 1, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
This question review and source binding
Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.