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Convergence Functions

Convergence of functions is introduced in Sequence of Functions | Real Analysis.

Uniform Convergence Theorem

Let be a sequence of Riemann integrable functions on . Suppose converges to uniformly. Then is Riemann integrable on and :

and:

When converges to pointwise, we cannot be sure if is Riemann integrable or not. An example where is not Riemann integrable:

Here is the enumeration of rational numbers in .

Dominated Convergence Theorem

Let be a sequence of Riemann integrable functions on . Suppose converges to pointwise where is Riemann integrable on . If :

Monotone Convergence Theorem

Let be a sequence of Riemann integrable functions on , and they are monotone (all increasing or decreasing, like ). Suppose converges to pointwise where is Riemann integrable on . If :

Can be proven from the dominated convergence theorem.