one-sided continuity by series
Theorem. If the function series
Proof. Suppose that the terms are continuous from the right. Let be any positive number and
Let now and with . Since every is continuous from the right in , the same is true for the finite sum , and therefore there exists a number such that
Thus we obtain that
as soon as
This means that is continuous from the right in an arbitrary point of .
Analogously, one can prove the assertion concerning the continuity from the left.
|Title||one-sided continuity by series|
|Date of creation||2013-03-22 18:34:03|
|Last modified on||2013-03-22 18:34:03|
|Last modified by||pahio (2872)|
|Synonym||one-sided continuity of series with terms one-sidedly continuous|