A sequence or function is said to be monotonic if it is

• monotonically increasing
• monotonically decreasing
• monotonically nondecreasing
• monotonically nonincreasing

Intuitively, this means that a monotone sequence can be thought of a ``staircase'' going either only up, or only down, with the stairs any height and any depth. The same goes for step functions. Smooth (analytic) monotone functions are like ``slopes,'' either up or down at potentially varying steepnesess.