Math 2.0

Real functions Counterexample

Dirichlet function

Builds on Continuity at a pointSequential continuity

Counterexample

Let f ⁣:ℝ→ℝf \colon \mathbb{R} \to \mathbb{R} be given by f(x)=1f(x) = 1 for rational xx and f(x)=0f(x) = 0 for irrational xx. Then ff is not continuous at any point. Indeed, each real aa is the limit of a sequence of rationals and of a sequence of irrationals. Their images tend to 11 and to 00, so one of them does not tend to f(a)f(a).

Remarks

The function ff is the indicator function of ℚ\mathbb{Q}. Every rational number is a period of ff. The function is not Riemann integrable on any interval with more than one point.

Reading path

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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 sequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionSequential continuity, definition, no evidence levelSequential continuitydefinitionDirichlet function, counterexample, no evidence levelDirichlet functioncounterexample
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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. Continuity at a point definition
  6. Limit of a sequence definition
  7. Sequential continuity definition
  8. Dirichlet function counterexample

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Dirichlet_function, Topological properties
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