Math 2.0

statement corollary

Continuous image of a closed interval is a closed interval

Statement

Let a≤ba \le b and let f ⁣:[a,b]→Rf \colon [a, b] \to \mathbb{R} be continuous. Then f([a,b])=[m,M]f([a, b]) = [m, M], where mm and MM are the minimum and the maximum of ff on [a,b][a, b].

Remarks

The extreme value theorem gives mm and MM as values of ff. The intermediate value theorem gives all values in between. The image can be a single point, when ff is constant. Mathlib states it as ContinuousOn.image_Icc.

Explanations

L1 human

A continuous function maps a closed bounded interval onto a closed bounded interval. The graph reaches a lowest and a highest point, and it takes every height in between.

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To read this node, read these 18 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 levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSubsequence, definition, no evidence levelSubsequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionBounded 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 functiondefinitionConvergent sequence, definition, no evidence levelConvergent sequencedefinitionMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementEvery sequence has a monotone subsequence, statement, no evidence levelEvery sequence has a …statementIntermediate value theorem, statement, no evidence levelIntermediate value th…statementBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementBoundedness theorem, statement, no evidence levelBoundedness theoremstatementExtreme value theorem, statement, no evidence levelExtreme value theoremstatementContinuous image of a closed interval is a closed interval, statement, no evidence levelContinuous image of a…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. Limit of a function definition
  4. Real sequence definition
  5. Subsequence definition
  6. Continuity at a point definition
  7. Bounded sequence definition
  8. Limit of a sequence definition
  9. Monotone sequence definition
  10. Continuous function definition
  11. Convergent sequence definition
  12. Monotone convergence theorem for sequences statement
  13. Every sequence has a monotone subsequence statement
  14. Intermediate value theorem statement
  15. Bolzano–Weierstrass theorem statement
  16. Boundedness theorem statement
  17. Extreme value theorem statement
  18. Continuous image of a closed interval is a closed interval statement

Sources

  1. Web https://en.wikipedia.org/wiki/Intermediate_value_theorem, Theorem

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