Question

What is the limit of the greatest integer function?

Answer 1

See explanation...

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.