Point to point convergence / Uniform convergence

[Tolaso J Kos]

Does the sequence of functions \( \displaystyle f_n(x)=\frac{x^{2n}}{1+x^{2n}}, \; n=1, 2, \dots , x \in \mathbb{R} \) converge point to point?

Is the convergence uniform?

[Grigorios Kostakos]

Easy to check that \[f_{n}(x)\xrightarrow{pointwise}f(x)=
\begin{cases}
1\,, & x\in(-\infty,-1)\cup(1,+\infty)\\
0\,, & x\in(-1,1)\\
\frac{1}{2}\,, & x=\pm1
\end{cases}\,.\] Because the functions \(f_n(x)\) are continuous and the function \(f\) is not continuous, the sequence \(\{{f_n}\}_{n\in\mathbb{N}}\) does not converges uniformly to \(f\).