Math 2.0

statement theorem

One-sided limits of a monotone function exist

Statement

Let a<c<ba < c < b and let f ⁣:(a,b)→Rf \colon (a, b) \to \mathbb{R} be increasing. Then both one-sided limits of ff at cc exist, and lim⁡x→c−f(x)=sup⁡a<x<cf(x)≤f(c)≤inf⁡c<x<bf(x)=lim⁡x→c+f(x).\lim_{x \to c^-} f(x) = \sup_{a < x < c} f(x) \le f(c) \le \inf_{c < x < b} f(x) = \lim_{x \to c^+} f(x).

Remarks

The proof is the same as for the monotone convergence theorem. The supremum exists because f(c)f(c) is an upper bound. A similar statement holds for decreasing functions, with the inequalities reversed. So ff is continuous at cc exactly when the two one-sided limits are equal. Otherwise cc is a jump discontinuity. Mathlib states the left limit as Monotone.tendsto_nhdsLT.

Explanations

L1 human

An increasing function can only move up as xx approaches cc from the left. Its values are bounded by f(c)f(c), so they settle at a limit.

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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 sequencedefinitionOne-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 sequencedefinitionMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementOne-sided limits of a monotone function exist, statement, no evidence levelOne-sided limits of a…statement
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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. One-sided limit definition
  6. Bounded sequence definition
  7. Limit of a sequence definition
  8. Monotone sequence definition
  9. Monotone convergence theorem for sequences statement
  10. One-sided limits of a monotone function exist statement

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Sources

  1. Web https://en.wikipedia.org/wiki/Monotonic_function, Some basic applications and results

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