\begin{align*} \limsup x_{n}=\lim _{N \rightarrow \infty} \sup _{n>N} x_{n} .\end{align*}
- For \((x_n) \coloneqq\qty{n+1\over n}\), \(\sup_n x_n = 2\) but \(\limsup_n x_n = 1\).
A nowhere differentiable function:
Monotone functions are differentiable almost everywhere.
Can there be a function \(f:I\to {\mathbf{R}}\) that is continuous on \({\mathbf{Q}}\) and discontinuous otherwise?