## Counterexamples on Uniform Convergence

List of Examples

Chapter 1. Conditions of Uniform Convergence

Example 1 . A function f ( x , y ) defined on converges pointwise on X as y approaches y 0, but this convergence is nonuniform on X

A sequence of functions converges (pointwise) on a set, but this convergence is nonuniform.

A series of functions converges (pointwise) on a set, but this convergence is nonuniform.

Example 2 . A series of functions converges on X and a general term of the series converges to zero uniformly on X , but the series converges nonuniformly on X .

Example 3 . A sequence of functions converges on X and there exists its subsequence that converges uniformly on X , but the original sequence does not converge uniformly on X .

Example 4 . A function f ( x , y ) defined on converges on ( a , b ) as y approaches y 0 and this convergence is uniform on any interval , but the convergence is nonuniform on ( a , b ).

A sequence of functions defined on ( a , b ) converges uniformly on any interval , but the convergence is nonuniform on ( a , b )

A series of functions converges uniformly on any interval , but the convergence is nonuniform on ( a , b ).

Example 5 . A sequence converges on X , but this convergence is nonuniform on a closed interval .

A series converges on X , but this series does not converge uniformly on a closed subinterval .

A function f ( x , y ) defined on has a limit for , but f ( x , y ) converges nonuniformly on a subinterval .

Example 6 . A sequence converges on a set X , but it does not converge uniformly on any subinterval of X .

A series converges on X , but it does not converge uniformly on any subinterval of X .

Example 7 . A series converges uniformly on an interval, but it does not converge absolutely on the same interval.

Example 8 . A series converges absolutely on an interval, but it does not converge uniformly on the same interval.

Example 9 . A series converges absolutely and uniformly on [ a , b ], but the series does not converge uniformly on [ a , b ].

Example 10 . A series converges absolutely and uniformly on X , but there is no bound of the general term u n ( x ) on X in the form , such that the series converges.

Example 11 . A sequence f n ( x ) converges uniformly on X to a function f ( x ), but does not converge uniformly on X to f 2( x ).

Sequences f n ( x ) and g n ( x ) converge uniformly on X , but f n ( x ) g n ( x ) does not converge uniformly on X .

Example 12 . Sequences f n ( x ) and g n ( x ) converge nonuniformly on X to f ( x ) and g ( x ), respectively, but converges to uniformly on X .

Example 13 . A sequence converges uniformly on X , but f n ( x ) diverges on X .

A sequence converges uniformly on X , but f n ( x ) diverges on X .

A sequence converges uniformly on X and f n ( x ) converges on X , but the convergence of f n ( x ) is nonuniform.

Example 14 . A sequence converges uniformly on X to 0, but neither f n ( x ) nor g n ( x ) converges to 0 on X .

Example 15 . A sequence f n ( x ) converges uniformly on X to a function f ( x ), , , , but do