limit function of sequence

Theorem 1.

Let  f1,f2,  be a sequence of real functions all defined in the interval[a,b].  This functionMathworldPlanetmath sequence converges uniformly to the limit function f on the interval  [a,b]  if and only if


If all functions fn are continuousMathworldPlanetmath in the interval  [a,b]  and  limnfn(x)=f(x)  in all points x of the interval, the limit function needs not to be continuous in this interval; example  fn(x)=sinnx  in  [0,π]:

Theorem 2.

If all the functions fn are continuous and the sequence  f1,f2,  converges uniformly to a function f in the interval  [a,b],  then the limit function f is continuous in this interval.

Note.  The notion of can be extended to the sequences of complex functions (the interval is replaced with some subset G of ).  The limit function of a uniformly convergent sequence of continuous functions is continuous in G.

Title limit function of sequence
Canonical name LimitFunctionOfSequence
Date of creation 2013-03-22 14:37:45
Last modified on 2013-03-22 14:37:45
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 22
Author pahio (2872)
Entry type Theorem
Classification msc 26A15
Classification msc 40A30
Related topic LimitOfAUniformlyConvergentSequenceOfContinuousFunctionsIsContinuous
Related topic PointPreventingUniformConvergence
Defines function sequence
Defines limit function