point preventing uniform convergence
Theorem. If the sequence of real functions converges at each point of the interval but does not converge uniformly on this interval, then there exists at least one point of the interval such that the function sequence converges uniformly on no closed sub-interval of containing .
Proof. Let the limit function of the sequence on the interval be . According the entry uniform convergence on union interval, the sequence can not converge uniformly to both half-intervals and , since otherwise it would do it on the union . Denote by the first (fom left) of those half-intervals on which the convergence is not uniform. We have . Then the interval is halved and chosen its half-interval on which the convergence is not uniform. We can continue similarly arbitrarily far and obtain a unique endless sequence
of nested intervals on which the convergence of the function sequence is not uniform, and besides the length of the intervals tend to zero:
This means that not uniformly on , whence the function sequence does not converge uniformly on the arbitrarily chosen subinterval of containing .
|Title||point preventing uniform convergence|
|Date of creation||2013-03-22 17:27:18|
|Last modified on||2013-03-22 17:27:18|
|Last modified by||pahio (2872)|