# Derivative of $\lceil 1/x \rceil$

I'm looking for the derivative $\frac{d}{dx}\lceil 1/x \rceil$.

I would like to find a real number $1<x \le y$ satisfying the minimum of $\left\lceil \frac{y}{x} \right\rceil x$, when $y$ is a fixed value $>0$.

Is the ceiling function a problem ?

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Thank you. $x$ can be taken $\ge 1$. If this ceiling function is a problem, is there another method to determine a minimum ? –  Dingo13 Oct 14 '13 at 22:41
Sorry I've forgotten a term in my problem. Thank you again for the help. –  Dingo13 Oct 14 '13 at 22:43
Ah, that makes a huge difference. I would drop your first and last sentences and just state the real problem: derivative won’t help. –  Brian M. Scott Oct 14 '13 at 22:44
The problem was indeed easy without this missing term. –  Dingo13 Oct 14 '13 at 22:45
Maybe there is a method to approximate correctly for the argument of minimum, for instance by bounding the ceiling function. No ? If for instance I look the derivative for the upper bound ? –  Dingo13 Oct 14 '13 at 22:48

Let $f(x)=\left\lceil\frac{y}x\right\rceil x$. If $x>y$, then $f(x)=x$, so $f(x)$ can be made arbitrarily large.

If we limit $x$ to the interval $[1,y]$, however, things are more interesting. Let $n=\lceil y\rceil$; as $x$ runs from $1$ to $y$, $\left\lceil\frac{y}x\right\rceil$ runs from $n$ down to $1$. Let $x_n=1$, and for $k=1,2,\ldots,n-1$ let $x_k=\frac{y}k$; then

$$\left\{x\in[1,y]:\left\lceil\frac{y}x\right\rceil=k\right\}=[x_k,x_{k-1})\;.$$

For $2\le k\le n$ the function $f(x)$ is increasing on $[x_k,x_{k-1})$, and

$$\lim_{x\to x_{k-1}^-}f(x)=kx_{k-1}=\frac{ky}{k-1}\;,$$

though this limit is not attained. Finally, $\frac{k}{k-1}$ is maximized at $k=2$, so as $x$ approaches $x_1=y$ from the left, $f(x)$ approaches $2y$ from below. The function $f(x)$ does not actually attain a maximum on $[1,y]$, but it is bounded above by $2y$ and attains values arbitrarily close to this.

Added: Somehow I managed to read minimum as maximum. Of course the analysis above shows that the minimum on $[x_k,x_{k-1})$ is $$f(x_k)=k\frac{y}{x_k}=y\;,$$ so the function achieves a minimum value of $y$ at each $x_k$.

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Thank you, but in fact I'm looking for a minimum, not a maximum. –  Dingo13 Oct 15 '13 at 9:06
I'm looking for the minimum when $1<x\le y$. –  Dingo13 Oct 15 '13 at 9:50
@Dingo13: That information was already available in what I’d written, but I’ve made it explicit and corrected some errors resulting from trying to type on a Kindle. –  Brian M. Scott Oct 15 '13 at 10:15

$\large x > 0:$

\begin{align} \left\lceil x\right\rceil &= \Theta\left(x\right)\Theta\left(1 - x\right) + 2\Theta\left(x - 1\right)\Theta\left(2 - x\right) + 3\Theta\left(x - 2\right)\Theta\left(3 - x\right) + \cdots \\[3mm]&= \sum_{n = 0}^{\infty}\left(n + 1\right)\Theta\left(x - n\right) \Theta\left(n + 1 - x\right) \end{align}
\begin{align} {{\rm d}\left\lceil x\right\rceil \over {\rm d}x} &= \sum_{n = 0}^{\infty}\left(n + 1\right)\left[% \delta\left(x - n\right) \Theta\left(n + 1 - x\right) - \Theta\left(x - n\right) \delta\left(n + 1 - x\right) \right] \\[3mm]&= \sum_{n = 0}^{\infty}\left(n + 1\right)\left[% \delta\left(x - n\right) \Theta\left(1\right) - \Theta\left(1\right) \delta\left(n + 1 - x\right) \right] \\[3mm]&= \sum_{n = 0}^{\infty}\left(n + 1\right) \delta\left(x - n\right) - \sum_{n = 1}^{\infty} n\,\delta\left(n - x\right) = \sum_{n = 0}^{\infty}\delta\left(x - n\right) \end{align}
\begin{align} {{\rm d}\left\lceil x\right\rceil \over {\rm d}x} &= \sum_{n = 0}^{\infty}\delta\left(x - n\right)\,, \qquad \left\lceil x \right\rceil = 1\ \mbox{when}\ 0 < x < 1 \end{align}
\begin{align} {{\rm d}\left\lceil 1/x\right\rceil \over {\rm d}x} &= \sum_{n = 0}^{\infty}\delta\left({1 \over x} - n\right)\, \left(-\,{1 \over x^{2}}\right) = \sum_{n = 1}^{\infty} {\delta\left(x - 1/n\right) \over \left\vert -1/x^{2}\right\vert}\, \left(-\,{1 \over x^{2}}\right) \end{align}
\begin{align} \color{#ff0000}{\large{{\rm d}\left\lceil 1/x\right\rceil \over {\rm d}x}} &\color{#000000}{\large\ =\ } \color{#ff0000}{\large -\sum_{n = 1}^{\infty}\delta\left(x - {1 \over n}\right)} \end{align}
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Thank you for the derivative. I suppose that $\delta$ is the Dirac function which is valued at 1 for 0 and 0 otherwise. As Brian has said, this derivative does not help a lot. –  Dingo13 Oct 15 '13 at 9:08