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The floor of a real number is the greatest integer less than or equal to the number. The floor of $x$ is usually denoted by $\lfloor x\rfloor$.
The real function $x\mapsto\lfloor{x}\rfloor$ is monotonically nondecreasing and satisfies
$x1<\lfloor{x}\rfloor\leqq x$ 
for all $x$. The function is continuous everywhere except in the integer points $0,\,\pm 1,\,\pm 2,\,\ldots$ where it is only continuous from the right. One has
$\lfloor\lfloor{x}\rfloor\rfloor\;=\;\lfloor{x}\rfloor,$ 
i.e. the function is idempotent.
Some examples:

$\lfloor 6.2\rfloor=6$,

$\lfloor 0.4\rfloor=0$,

$\lfloor 7\rfloor=7$,

$\lfloor5.1\rfloor=6$,

$\lfloor\pi\rfloor=3$,

$\lfloor4\rfloor=4$.
Note that this function is not the integer part ($[x]$), since $\lfloor3.5\rfloor=4$ and $[3.5]=3$. However, both functions agree for nonnegative numbers.
The notation for floor and ceiling was introduced by Iverson in 1962[1]. In some texts however, the bracket notation is used to denote the floor function (although they actually work with integer part) so it is sometimes also called the bracket function.
References
 1 N. Higham, Handbook of writing for the mathematical sciences, Society for Industrial and Applied Mathematics, 1998.
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