Examples and counter-examples in analysis

Vincent Pantaloni, Aug 4, 2019

Animations to illustrate lessons in analysis

Table of Contents

  1. Pointwise convergence
    1. Boundedness is not preserved by pointwise convergence.
    2. Continuity is not preserved by pointwise convergence.
    3. Differentiability is not preserved by pointwise convergence.
    4. fn converges does not imply f'n converges
    5. Integration counter-examples : Witch's hat functions
    6. Integration and uniform convergence

1.

Pointwise convergence

Some animations to illustrate what pointwise convergence is and some counter-examples regarding this notion.

  • 1. Boundedness is not preserved by pointwise convergence.
  • 2. Continuity is not preserved by pointwise convergence.
  • 3. Differentiability is not preserved by pointwise convergence.
  • 4. fn converges does not imply f'n converges
  • 5. Integration counter-examples : Witch's hat functions
  • 6. Integration and uniform convergence

1.1.

Boundedness is not preserved by pointwise convergence.

Each f_n is bounded (by n) on [0,1], the sequence f_n converges pointwise to f on ]0,1], but f is not bounded on ]0,1]. [br]This counter-example shows that pointwise convergence does not, in general, preserve boundedness.
GIF animation
Each f_n is bounded (by n) on [0,1], the sequence f_n converges pointwise to f on ]0,1], but f is not bounded on ]0,1].
GeoGebra applet
– Vincent Pantaloni

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1.5.

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1.6.

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