Continuity of the integer part

Statement

The “integer part” function $\lfloor x\rfloor$ is piecewise-defined (on each $[n,n+1)$ it equals $n$): is it continuous? At which points does it have jumps?

Solution

On each interval $[n,n+1)$ the function is constant and equals $n$, so it is continuous inside each piece. Problems arise at the boundaries, that is, at each integer $n$: $\lim_{x\to n^-}\lfloor x\rfloor = n-1, \qquad \lfloor n\rfloor = n, \qquad \lim_{x\to n^+}\lfloor x\rfloor = n.$ The left-hand limit ($n-1$) does not coincide with the value ($n$): there is a jump of height $1$ at every integer.

Conclusion: $\lfloor x\rfloor$ is continuous at every non-integer point and has a jump at every point of $\mathbb{Z}$ (a “step” function).

Links

Topics: Functions and properties
Concepts: Continuity · Piecewise-defined function · Integer part
Functions: Integer part
Skills: Reasoning by cases
Exercise type: Function study