Math 2.0

statement theorem

Continuity of the inverse of a strictly monotone continuous function

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.

Explanations

L1 human

The graph of f−1f^{-1} is the graph of ff reflected in the line y=xy = x. A rising unbroken graph stays rising and unbroken after the reflection.

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To read this node, read these 15 nodes in this order. The foundations come first, and the list ends with this node.

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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
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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

What depends on this

22 nodes use this node, directly or through other nodes.

Sources

  1. Web https://en.wikipedia.org/wiki/Inverse_function, Standard inverse functions

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