one-sided continuity by series

Theorem.  If the function series

n=1fn(x) (1)

is uniformly convergent on the interval[a,b],  on which the fn(x) are continuousMathworldPlanetmath from the right or from the left, then the sum function S(x) of the series has the same property.

Proof.  Suppose that the terms fn(x) are continuous from the right.  Let ε be any positive number and


where Sn(x) is the nth partial sum of (1) (n= 1, 2,).  The uniform convergence implies the existence of a number nε such that on the whole interval we have


Let now  n>nε  and  x0,x0+h[a,b]  with  h>0.  Since every fn(x) is continuous from the right in x0, the same is true for the finite sum Sn(x), and therefore there exists a number δε such that

|Sn(x0+h)-Sn(x0)|<ε3when  0<h<δε.

Thus we obtain that

|S(x0+h)-S(x0)| =|[Sn(x0+h)-Sn(x0)]+Rn+1(x0+h)-Rn+1(x0|

as soon as


This means that S is continuous from the right in an arbitrary point x0 of  [a,b].

Analogously, one can prove the assertion concerning the continuity from the left.

Title one-sided continuity by series
Canonical name OnesidedContinuityBySeries
Date of creation 2013-03-22 18:34:03
Last modified on 2013-03-22 18:34:03
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Theorem
Classification msc 40A30
Classification msc 26A03
Synonym one-sided continuity of series with terms one-sidedly continuous
Related topic OneSidedContinuity
Related topic SumFunctionOfSeries