What is the derivative of the floor function?

Question

What is the derivative of the following equation?

$$f(x) = \left \lfloor \frac{c}{x} \right \rfloor$$

• $c,x$ are positive integers, and $\lfloor \cdot \rfloor$ is the floor function.

Does the floor function play any role here? Will it be equal to the floor of the derivative?

Answer 1

$f'(x) = \begin{cases} \text{not differentiable}&\text{if } x=\frac cn \text{ for some positive integer }n, \\ 0&\text{othewise.}\end{cases}$

Hint: Just do it by the definition of the derivative. (It is not difficult, so I omit the proof.)

Answer 2

As Syuizen points out, $\lfloor x \rfloor$ is not differentiable as a function.

However, as a distribution, you can write the derivative of $\lfloor x \rfloor$ as an infinite sum of Dirac delta functions:

$$\frac{d}{dx} \lfloor x \rfloor = \sum_{n \in \mathbb Z} \delta_n(x).$$