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# 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.

Example:

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.