Q:

Type of discontinuities in Dirichlet function

I notice three types of discontinuities "removable", "jump", "infinite" defined here Classification of Discontinuities involve limit.  Then I am puzzled that what is the type of discontinuities in the Dirichlet function, in which all rational numbers are mapped to 1 and irrational numbers are mapped to 0?  They seem not to fall in any of these three categories?  Thank you. 

A:

The classification you link to defines an "infinite discontinuity" to mean the case where "one or both of the limits $L^{-}$ and $L^{+}$ does not exist or is infinite". 
That's precisely what happens with the Dirichlet function everywhere.