Pointwise and Uniform Convergence
Definition: Letbe a subset ofand letbe a sequence of functions defined onWe say thatconverges pointwise onif exists for each point
This means thatis a real number that depends only on
Ifis pointwise convergent then the function defined by
for everyis called the pointwise limit of the sequence
More formally, given anyand there exists a natural numberdepending only onandsuch that
Definition Letbe a subset ofand letbe a sequence of real valued functions defined onThenconverges uniformly toif given anythere exists a natural numbersuch thatfor everydepends only on so uniform convergence implies pointwise convergence., but the converse is not true.
Asfor eachso converges pointwise to zero.
Suppose now thatandthenandsois not uniformly convergent.