Question

For any real number x there is a unique integer n such that n≤x<n+1, and the greatest integer function is defined as ⌊x⌋=n. Where are the critical values of the greatest integer function? Which are local maxima and which are local minima?

Answer 1

⌊x⌋ has no maxima and minima. It is a piecewise linear function, and each piece is a horizontal segment of length 1.

Since a critical point is where the derivative is zero or undefined, then that would be every real number, since

at the integers, ⌊x⌋ is discontinuous, so the derivative is undefined.

elsewhere, the derivative is zero.