# Is the function $\sum_{n=0}^{\infty} \frac{n^2x}{1+n^4x^2}$ continuous in $x \in (0, 1]$?

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.

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## 1 Answer

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

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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 '14 at 12:35