Dirichlet’s function is nowhere continuous and nowhere differentiable. It is also nowhere Riemann integrable since its upper integral and lower integral do not equal anywhere.
Read & learn:
http://math.feld.cvut.cz/mt/txtb/4/txe3ba4s.htm
Reference:
http://mathworld.wolfram.com/DirichletFunction.html
#math #science #analysis #calculus
Wow, nowhere continuous. That comes as a surprise. I would have thought there were small stretches of R that don't have intervening elements from Q and so small lengths of continuity for this function.
ReplyDeleteBring it on! ;)
ReplyDeleteSounds like the boundary condition for quantum entanglement to me.
ReplyDeleteIt's like a variation of Zeno's Paradox. In this case though, between any two points in R/Q, there exists a point in Q and vice versa.
ReplyDeleteBringing back sweet ol' memories. Sleeping in the last bench in math class :) Then I became an engineer and only after finishing school I started to understand what they were talking about. And I couldn't get enough of.
ReplyDeleteLooks like an AM modulation spectrum to me.
Thanks.