Basic properties of the floor function |_ x _|:

Problem A:

(1)  |_x_| + |_y_| <= |_x + y_| <= |_x_| + |_y_| + 1

(2)  |_2x_| + |_2y_| >= |_x + y_| + |_x_| + |_y_|


Problem B: [Knu68, 1.2.4 Exc 34][GKP88, 3.2 (3.10)]
 Let f:A->R be a strictly increasing continuous function in the reel numbers.
 Where A is a subset of R with the property x in A => |_x_| in A and |¯x¯| in A.
 Then the following conditions are equivalent
 (a)  f(x) in Z  =>  x in Z
 (b)  |_f(x)_|  =  |_f(|_x_|)_|
 (c)  |¯f(x)¯|  =  |¯f(|¯x¯|)¯|



Solution A:
                                                      _ _           _ _
(1)  |_x_| + |_y_| = |_|_x_| + y_| <= |_x + y_| <= |_| x | + y_| = | x | + |_y_|
  
(2)  We only need to show the case 0<=x<1, 0<=y<1. Let x >= y then 2x >= x+y.
     Therefore |_2x_| >= |_x + y_|. As |_x_| = |_y_| = 0 we derive 
     |_2x_| + |_2y_| >= |_2x_| >= |_x + y_| = |_x + y_| + |_x_| + |_y_|.



References:

Eng98: Arthur Engel;
   Problem-solving strategies,
   Problem Books in Mathematics. New York, NY: Springer. x, 403 p. (1998)
   ISBN 0-387-98219-1/hbk
   Zbl 887.00002
   Chap 14.6 Integer Function 

GKP88: Ronald L. Graham, Donald E. Knuth, Oren Pataschnik;
   Concrete Mathematics,
   Addison Wessley, Reading (1994) 2nd Ed.
   Chap 3: Integer Functions
   Sect 3.1: Floors and Ceilings
   Sect 3.2: Floor/Ceiling Applications
   Sect 3.3: Floor/Ceiling Recurrences
   Sect 3.4: Mod: the Binary Operation
   Sect 3.5: Floor/Ceiling Sums

Ive62: K. E. Iverson, 
   A programming language, Wiley, 1962.
   (invents the notation |_  _|)

Knu68: Donald E. Knuth;
   The Art of Computer Programming Vol. 1,
   Fundamental Algorithms,
   Addison Wessley, Reading (1973) 2nd Ed.
   Chapter 1.2.4: Integer Functions and Elementary Number Theory
   Excercise 1.3.2.16-17 fixed precision harmonic series

NiZ60: Ivan Niven, Herbert S. Zuckerman;
   An Introduction to the Theory of Numbers,
   John Wiley, New York (1972) 3rd edition. (1st edition (1960))
   (german: Einf\"uhrung ind die Zahlentheorie I, BI 46 (1976))
   Chap 4: Functions in Number Theory
   Sect 4.1: The Function [x]

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