What is the limit of the greatest integer function?

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Answer 1 · 2016-03-22T23:58:24

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The "greatest integer" function otherwise known as the "floor" function has the following limits:

$lim_(x->+oo) floor(x) = +oo$

$lim_(x->-oo) floor(x) = -oo$

If $n$ is any integer (positive or negative) then:

$lim_(x->n^-) floor(x) = n-1$

$lim_(x->n^+) floor(x) = n$

So the left and right limits differ at any integer and the function is discontinuous there.

If $a$ is any Real number that is not an integer, then:

$lim_(x->a) floor(x) = floor(a)$

So the left and right limits agree at any other Real number and the function is continuous there.