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

Suppose is sequence of bounded functions. converges to uniformly iff:

Proof Hint

Let .

To prove :

• Consider the epsilon-delta definition of uniform convergence
• is an upperbound of

To prove :

• Consider the epsilon-delta definition of the above limit

Properties of uniform convergence

Continuity

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

Proof Hint

Consider the limit definitions of:

1. converges to
2. is continuous at

Consider . Introduce and in there. Split into 3 absolute values. Show that the sum is lesser than .

Integrability

Explained in Converging Functions | Riemann Integration.

Differentiability

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

Suppose is a sequence of differentiable functions, and they uniformly converges to . Differentiability of is not guaranteed. An example is:

is differentiable but is only differentiable on .

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