\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 \(\limsup_n x_n = 1\).
](2021-10-29_16-43-30.png){kind=link}
](2021-10-29_16-43-43.png){kind=link}
](2021-10-29_16-51-23.png){kind=link}
](2021-10-29_16-55-00.png){kind=link}
A nowhere differentiable function:
](2021-10-29_16-57-12.png){kind=link}
Monotone functions are differentiable almost everywhere.
](2021-10-29_17-28-39.png){kind=link}
Can there be a function \(f:I\to {\mathbb{R}}\) that is continuous on \({\mathbb{Q}}\) and discontinuous otherwise?
](2021-10-29_18-23-37.png){kind=link}