Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

We know that the function $\sum_{n=0}^{\infty} \frac{n^2x}{1+n^4x^2}$ is convergent but not uniformly convergent when $x$ is in $(0, 1]$. How do we know if it's continuous or not, though?

Help much appreciated.

share|improve this question

1 Answer 1

up vote 5 down vote accepted

Perhaps show it is uniformly convergent on $[a,1]$ for every $a>0$.

share|improve this answer
    
After showing that and deducing that it is continuous on [a,1], how do we make the logical leap to it being continuous on (0,1]? I'm tempted to just state it, but we know that uniform convergence on [a,1] does not imply uniform convergence on (0,1], so there must be some nuisances? –  dhz Nov 30 '11 at 1:30
    
Continuous on a set means continuous at each point of the set. Given some point $u \in (0,1]$, show your sum is continuous at $u$. –  GEdgar Nov 30 '11 at 4:36
    
Here is link to question asking precisely about the uniform convergence on such intervals: math.stackexchange.com/questions/678513/… –  Martin Sleziak Feb 17 at 12:35

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.