Class Contents

• Definition of uniform convergence
• Uniform limits of continuous functions are continuous
• Uniform limits of Riemann integrable functions

Remark:

Thus, uniform limits are unique.

Example:

Example:

The first example illustrates a useful test:

Example:

The best application of uniform convergence is to construct new functions from approximating sequences. Pointwise convergence does this too, but uniform limits inherit more of the useful properties of the approximating functions.

Proof.

Example:

Proof.

Remark:

In order to use that last result in a nontrivial way, we need to be able to recognize uniformly converging sequences when we don't know the limits.

Proof.

Analysis WebNotes by John Lindsay Orr.
Comments to the author: jorr@math.unl.edu

All contents copyright (C) 1996 John L. Orr
University of Nebraska--Lincoln
All rights reserved

Last modified: June 1996