If I still havent answered your question At what fractional does the fractional derivative does the nonsmoothness make the fractional derivative locally undefined  then let me just say that its the same as nonfractional integration  Youre really not doing anything different spatially speaking  And the formalism only describes whats going on spatially so if you try to measure the volume of a surface or something like that youll have problems  Just so long as you realize what youre actually doing there should be no surprises  And one more thing notice that integration has a region of integration but differentiation doesnt  Thats why you have to add that C back in when you reverse it  Same holds true for fractional integration only the C might be a function of fractional dimension  NASA invented an initialized fractional calculus to deal with this  You might find it interesting