Darboux's law of intermediate values
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Darboux's law of intermediate values is a fundamental theorem in real analysis stating that the image of a continuous function on an interval contains every value between any two of its function values.
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in real analysis ⓘ |
| alsoKnownAs |
Darboux property
NERFINISHED
ⓘ
Darboux's theorem (real analysis) NERFINISHED ⓘ |
| appliesTo |
continuous functions with values in R
ⓘ
functions defined on intervals of the real line ⓘ real-valued functions ⓘ |
| assumes |
domain is an interval in R
ⓘ
function is continuous on the interval ⓘ |
| characterizes | the image of a continuous function on an interval as connected ⓘ |
| conclusion | the range of the function on the interval is an interval ⓘ |
| contrastsWith | behavior of discontinuous functions that may have jump discontinuities ⓘ |
| coreStatement |
If f is continuous on an interval I and a,b are in I, then for every value y between f(a) and f(b) there exists c in I between a and b such that f(c)=y.
ⓘ
The image of a continuous function on an interval is an interval. ⓘ |
| ensures | no jump discontinuities for continuous functions on intervals ⓘ |
| field | real analysis ⓘ |
| formalizes | intuitive idea that graphs of continuous functions on intervals have no gaps in their vertical values ⓘ |
| generalizationOf | the idea that continuous functions on intervals cannot skip values ⓘ |
| hasProperty |
does not require differentiability
ⓘ
non-constructive in typical proofs ⓘ |
| historicalPeriod | 19th-century mathematics ⓘ |
| holdsIn | standard real number system R ⓘ |
| implies |
continuous real functions on intervals have the intermediate value property
ⓘ
images of intervals under continuous maps are intervals ⓘ intermediate value property for continuous functions ⓘ |
| importance |
basic tool in elementary real analysis
ⓘ
fundamental theorem in the foundations of calculus ⓘ |
| involves |
intervals of real numbers
ⓘ
order topology on R ⓘ |
| mathematicalDomain | analysis ⓘ |
| namedAfter | Jean Gaston Darboux NERFINISHED ⓘ |
| relatedConcept |
Darboux functions
NERFINISHED
ⓘ
connectedness of subsets of R ⓘ continuity ⓘ intermediate value property ⓘ |
| relatedTo |
Bolzano's theorem
NERFINISHED
ⓘ
existence of roots of continuous functions on intervals ⓘ intermediate value theorem ⓘ |
| requires |
completeness of the real numbers for standard proofs
ⓘ
order structure of the real numbers ⓘ |
| topicOf | university-level analysis courses ⓘ |
| usedIn |
proofs of the intermediate value theorem
ⓘ
real analysis textbooks ⓘ theory of differential equations ⓘ topology of the real line ⓘ |
| usedToShow | existence of solutions to equations f(x)=y for intermediate values y ⓘ |
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.