From: PointedEars@web.de   
      
   Thomas 'PointedEars' Lahn wrote:   
   > One must distinguish between a function that is _identically_ zero, i.e.   
   > whose value is zero _everywhere_, and a function whose value is zero _for a   
   > finite number of arguments in its domain_.   
   >  > The derivative of the former function *is* actually zero because it is a   
   > special case of a constant function, but the derivative of the latter   
   > function is not necessarily zero.   
   Actually, one has to be even more careful with one's wording.   
      
   As we can see from periodic functions like the sine function, it is even   
   possible that a function is zero for a countably infinite number of   
   arguments (e.g. all integer multiples of π) but still not all arguments.   
      
   And one can even think of a pathological case: The Dirichlet function   
      
     1_ℚ(x) = {1 if x ∈ ℚ;   
               0 if x ∉ ℚ   
      
   is zero for *uncountably* infinitely many arguments in its domain because   
   they are real numbers but not rational numbers, and non-zero for *countably*   
   infinitely made arguments in its domain because they are rational numbers   
   (the latter are members of a countably infinite set, as Cantor proved).   
      
   So I should have said "a function whose value is zero for at least one   
   argument in its domain, but whose value is NOT zero for at least one other   
   argument in its domain".   
      
   With that said, the Dirichlet function is not continuous (a textbook proof   
   that can be made using the ε–δ criterion of continuity, often asked for in   
   mathematics exercises and exams -- even physics students have to do those;   
   BTDT), therefore not differentiable, and it is moot to ask about its   
   derivative because that does not exist.   
      
      
      
   --   
   PointedEars   
      
   Twitter: @PointedEars2   
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