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Greatest integer function (floor value function)

y = f(x) = [x] (Greatest integer ≤ x)
 
Domain R; Range I;
 
78054.png

Properties of greatest integer function

  • [x] = n (n I) x [n, n +1)
  • x – 1 < [x] ≤ x
  • [–x] + [x] = 78047.png
  • [x] ≥ n x n, n I
  • [x] ≤ n x < n + 1, n I
  • [x] > n xn + 1, n I
     
    For example,
     
    [x] ≥ 2 x [2, ∞)
     
    [x] > 3 [x] ≥ 4 x [4, ∞)
     
    [x] ≤ 3 x (–∞, 4)
Fractional part function
 
y = f(x) = {x} = x – [x]
 
Domain R; Range [0, 1); Period 1;
 
[x + y] = [x] + [y], if 0 ≤ {x} + {y} < 1
 
[x + y] = [x] + [y] + 1 , 1 ≤ {x} + {y} < 2
 
{x} + {–x} = 0 if x I
 
{x} + {–x} = 1 if x I
 
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