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Sequence of Functions

Types of Convergence

Pointwise convergence

Here depends on .

Examples:

  • on

Uniformly convergence

Here depends on only. Implies pointwise convergence.

Examples:

  • on

Uniform convergence tests

Supremum test

A sequence of functions converges to uniformly iff:

Properties of uniform convergence

Continuity is preserved

If is continuous and converging to , then is also continuous.

Limit and integral can be switched

Explained in Converging Functions | Riemann Integration.

Differentiation is complicated

Uniform convergence-differentiation pair doesn’t go as smooth like integration was.

Suppose is a sequence of differentiable functions, and they uniformly converges to . Then we can’t say, for sure, is differentiable. An example is:

Theorem

If (all conditions must be met):

  1. is differentiable on
  2. converges (pointwise) for some
  3. converges to uniformly on

Then:

  1. converges to uniformly on
  2. is differentiable on
  3. OR in other words converges to uniformly