definition (limsup):
lim sup
example (?):
- For (x_n) \coloneqq\qty{n+1\over n}, \sup_n x_n = 2 but \limsup_n x_n = 1.
theorem (Bounded Convergence):
example (?):
example (A nowhere differentiable function):
A nowhere differentiable function:
fact:
Monotone functions are differentiable almost everywhere.
exercise (?):
Can there be a function f:I\to {\mathbf{R}} that is continuous on {\mathbf{Q}} and discontinuous otherwise?
solution: