Math 2.0

Real functions Theorem

Continuity of the inverse of a strictly monotone continuous function

Builds on Strictly monotone functionIntermediate value theoremOne-sided limits of a monotone function exist

Statement

Let II be an interval and let f ⁣:I→ℝf \colon I \to \mathbb{R} be continuous and strictly increasing. Then J=f(I)J = f(I) is an interval, and the inverse function f−1 ⁣:J→If^{-1} \colon J \to I is continuous and strictly increasing.

Remarks

The set JJ is an interval by the intermediate value theorem. The inverse is strictly increasing and its image is the interval II. A monotone function whose image is an interval has no jumps, so f−1f^{-1} is continuous. The same holds for strictly decreasing functions. This gives the continuity of nn-th roots, of the logarithm, and of the inverse trigonometric functions.

Reading path

To understand this page from the ground up, read these 15 pages in order. The foundations come first.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionSequence, definition, no evidence levelSequencedefinitionStrictly monotone function, definition, no evidence levelStrictly monotone fun…definitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionReal sequence, definition, no evidence levelReal sequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionOne-sided limit, definition, no evidence levelOne-sided limitdefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementIntermediate value theorem, statement, no evidence levelIntermediate value th…statementOne-sided limits of a monotone function exist, statement, no evidence levelOne-sided limits of a…statementContinuity of the inverse of a strictly monotone continuous function, statement, no evidence levelContinuity of the inv…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. Sequence definition
  3. Strictly monotone function definition
  4. Limit of a function definition
  5. Real sequence definition
  6. Continuity at a point definition
  7. One-sided limit definition
  8. Bounded sequence definition
  9. Limit of a sequence definition
  10. Monotone sequence definition
  11. Continuous function definition
  12. Monotone convergence theorem for sequences statement
  13. Intermediate value theorem statement
  14. One-sided limits of a monotone function exist statement
  15. Continuity of the inverse of a strictly monotone continuous function statement

Where this leads

26 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Inverse_function, Standard inverse functions
About this page
Tag07BY47HG5S
URIm2://07BY47HG5S
Kindstatement, theorem
WritingW0 stub
MSC26A48
StatusThis copy of the site has no build data. Run m2 status 07BY47HG5S.
UsesStrictly monotone function, Intermediate value theorem, One-sided limits of a monotone function exist

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