Math 2.0

statement theorem

Continuous functions on a closed bounded interval are uniformly continuous

Statement

Let a≤ba \le b and let f ⁣:[a,b]→Rf \colon [a, b] \to \mathbb{R} be continuous. Then ff is uniformly continuous on [a,b][a, b].

Remarks

Suppose not. Then there is ε>0\varepsilon > 0 and points xn,ynx_n, y_n with ∣xn−yn∣<1/n|x_n - y_n| < 1/n and ∣f(xn)−f(yn)∣≥ε|f(x_n) - f(y_n)| \ge \varepsilon. A convergent subsequence of (xn)(x_n) leads to a contradiction with continuity. The interval must be closed and bounded. The function 1/x1/x is continuous on (0,1](0, 1] but not uniformly continuous there. The name honors Eduard Heine and Georg Cantor. Mathlib states it for compact sets, as IsCompact.uniformContinuousOn_of_continuous.

Explanations

L1 human

On a closed bounded interval, continuity at each point upgrades to uniform continuity. There is no room for the function to become steeper and steeper.

What this node needs for W1
  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a node is. They do not measure its truth.

What this needs

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 functiondefinitionEpsilon-delta definition of continuity, statement, no evidence levelEpsilon-delta definit…statementConvergent 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 …statementUniform continuity, definition, no evidence levelUniform continuitydefinitionBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementUniform continuity on a set, definition, no evidence levelUniform continuity on…definitionContinuous functions on a closed bounded interval are uniformly continuous, statement, no evidence levelContinuous functions …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.
  • no evidence level
  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. Epsilon-delta definition of continuity statement
  12. Convergent sequence definition
  13. Monotone convergence theorem for sequences statement
  14. Every sequence has a monotone subsequence statement
  15. Uniform continuity definition
  16. Bolzano–Weierstrass theorem statement
  17. Uniform continuity on a set definition
  18. Continuous functions on a closed bounded interval are uniformly continuous statement

What depends on this

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

Sources

  1. Web https://en.wikipedia.org/wiki/Heine–Cantor_theorem, lead section

What links here

Used by