Math 2.0

Real functions Theorem

A continuous injective function on an interval is strictly monotone

Builds on Intermediate value theoremStrictly monotone function

Statement

Let II be an interval and let f ⁣:I→ℝf \colon I \to \mathbb{R} be continuous and injective. Then ff is strictly monotone.

Remarks

Suppose x<y<zx < y < z in II with f(y)f(y) not between f(x)f(x) and f(z)f(z). Then the intermediate value theorem gives a value of ff taken twice, which contradicts injectivity. Continuity is needed. The function on [0,2][0, 2] with f(x)=xf(x) = x for x<1x < 1 and f(x)=3−xf(x) = 3 - x for x≥1x \ge 1 is injective but not monotone. Mathlib states a form for closed intervals as ContinuousOn.strictMonoOn_of_injOn_Icc.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionStrictly monotone function, definition, no evidence levelStrictly monotone fun…definitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionIntermediate value theorem, statement, no evidence levelIntermediate value th…statementA continuous injective function on an interval is strictly monotone, statement, no evidence levelA continuous injectiv…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Limit point of a set definition
  2. Strictly monotone function definition
  3. Limit of a function definition
  4. Continuity at a point definition
  5. Continuous function definition
  6. Intermediate value theorem statement
  7. A continuous injective function on an interval is strictly monotone statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Monotonic_function, Inverse of function
About this page
Tag0D6FCW80A0
URIm2://0D6FCW80A0
Kindstatement, theorem
WritingW0 stub
MSC26A48
StatusThis copy of the site has no build data. Run m2 status 0D6FCW80A0.
UsesIntermediate value theorem, Strictly monotone function

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