Define $$f(x)=\sum_{n=0}^{\infty} \sin^{2}(nx)/ (1+n^{2} x^{2})$$
It was previously shown that $f$ is uniformly continuous on $\Bbb R_{\ne 0}$, $f$ converges (absolutely) for all real $x$ and that $f$ is continuous on $\Bbb R_{\ne 0}$
Now, I am seeking verification for my proof that $f$ is discontinuous at zero.
Proof: Firstly observe that $f(0)=0$. I now use an $\epsilon$ - $\delta$ argument. Fix $\epsilon = \frac{\sin^{2}(1)}{2}$ . For any $\delta = \frac{1}{k}$, set $x=\frac{1}{2k}$, so that $|x|<\delta$.
Then clearly $f(\frac{1}{2k})=\sum_{n=0}^{\infty} \sin^{2}(\frac{n}{2k})/ (1+n^{2} (\frac{1}{2k})^{2})$ > $\epsilon$. If this is unclear, consider the $n= 2k$-th term.
Hence $f$ is discontinuous at zero, qed.