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# What is the limit of the greatest integer function?

See explanation...

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.