Distributional Derivative of Floor Function

Theorem
Let $\floor x$ be the floor function.

Let $\map {\operatorname {III}} x$ be the Dirac comb.

Then the distributional derivative of $\floor x$ is $\map {\operatorname {III}} 0$.

Proof
By definition:


 * $\floor x := \sup \set {m \in \Z: m \le x}$

Hence, $\forall m \in \Z : \forall x \in \openint m {m + 1}$ the floor function is constant.

Therefore:


 * $\forall m \in \Z : \forall x \in \openint m {m + 1} : \dfrac {\d \floor x} {\d x} = 0$

Every $x \in \Z$ is a discontinuity of $\floor x$.

Hence, the jump rule has to be applied to each such $x$.

Suppose $k \in \Z$.

Then:


 * $\ds \lim_{x \to k^+} \floor x = k$


 * $\ds \lim_{x \to k^-} \floor x = k - 1$

By the Jump Rule: