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QUESTION

# What is the limit of the greatest integer function?

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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.