Pointwise Convergence for Functions

Convergence does not have a single definition. We may say that a function converges to a limit functionbut it is not enough to sayWe must be more precise.

A sequence of functionsconverges pointwise to a limit functionon a setif, for eachWe relate this definition to the ordinary definition which states:

is continuous at a pointif for eachthere existssuch that ifthen

Pointwise convergence implies that we must makedependent on

For example,converges to the zero functionon

To prove thatis continuous atwe must findsuch thatimplies

with r<1.

Takeso thatimplies that ifthen

Though useful, pointwise convergence does not give a type of convergence strong enough for many purposes. In fact for the case above, if we extend the domain slightly to include 1, the sequencedoes not even converge to a continuous function, sinceifbutif