Undergraduate Review

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: