File:Pompeiu derivative and integral.svg

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Summary

Description

English: Monotonic function

g : [0, 1] \to [0, 1]

defined by normalizing

$G (y) = \sum_{n = 0}^{\infty} a_{n} \sqrt[3]{x - q_{n}}$ where $q_{n}$ is the enumeration of all rational numbers in $[0, 1]$ in lexicographic order by the denominator, then the numerator; and $a_{n} = 0.9 9^{n}$ . That is, $g (y) = \frac{G (y) - x_{0}}{x_{1} - x_{0}},$ where $x_{0} \approx - 74.62226$ and $x_{1} \approx 75.73286$ . The derivative of this function is $g^{'} (y) = \frac{1}{x_{1} - x_{0}} \sum_{n = 0}^{\infty} \frac{a_{n}}{3 \sqrt[3]{(x - q_{n})^{2}}},$ which is strictly positive and infinite at every rational value of $y$ . Therefore, $(g^{- 1})^{'} (x) = 1 / (g^{'} \circ g^{- 1}) (x)$ is a Pompeiu derivative: $g^{- 1} (x)$ is differentiable at every point, but is zero on a dense subset of $[0, 1]$ .

For purposes of the plot, this function was approximated by truncating to fractions with denominator $\leq 1000$ , which yields $N = 304193$ terms and an absolute error bound as $| δ g (y) | \leq \frac{1}{x_{1} - x_{0}} \sum_{n = N}^{\infty} 0.9 9^{n} = \frac{0.9 9^{N}}{(x_{1} - x_{0}) (1 - 0.99)} \approx 1.201246 \times 1 0^{- 1328},$ too small to handle with floating-point numbers. However, a large number of terms is necessary to compute the zeros of $(g^{- 1})^{'}$ : Every $x$ where $g^{- 1} (x)$ is rational has $(g^{- 1})^{'} (x) = 0$ . We calculated $g (y)$ and $g^{'} (y)$ for $l c m {1, 2, ..., 15} + 1 = 360361$ evenly spaced values of $y$ in $[0, 1]$ in order to capture a fairly even mix of zero and nonzero values of $(g^{- 1})^{'} (x)$ ; then the plotted $x$ -values are simply the values of $g (y)$ .

The top plot is

g^{- 1} (x)

, and the bottom is its Pompeiu derivative. For both plots, we have zoomed in on the graphs in the intervals

I_{1} \approx [0.4575870, 0.5178396]

and

I_{2} \approx [0.9180436, 1]

, illustrating the finer details of the plots.

Date

28 March 2026

Source

Own work

Author

LaundryPizza03

SVG development

The SVG code is valid.

This plot was created with Matplotlib.

Source code

Python code

Source code

import math
import numpy as np
import matplotlib as mpl
import matplotlib.pyplot as plt
from time import time

time0 = time()

# An enumeration of the rational numbers between 0 and 1 with denominator <= B
def enumq(B):
    a = 0
    b = 0
    N = 0
    qn = []
    while b <= B:
        if a == b:
            b += 1
            a = 0
        else:
            a += 1
        if math.gcd(a,b) > 1: continue
        else:
            qn.append(a/b)
            N += 1

    return np.array(qn)

r = 0.99
# A summable series
def a(n):
    return r**n

def log10remainder(n):
    return n*np.log10(r) - np.log10(1-r)

# Pompeiu derivative (not normalized)
@np.vectorize(otypes=(float,),excluded=(1,2))
def invpompeiu(y,an,qn):
    return np.sum(an * np.cbrt(y-qn))

@np.vectorize(otypes=(float,),excluded=(1,2))
def dinvpompeiu(y,an,qn):
    return np.inf if y in qn else np.sum(an / (3*np.cbrt(y-qn)**2))

# Normalize range to [0,1]
def normalize(x,X0,X1):
    return (x-X0)/(X1-X0)

def dnormalize(x,X0,X1):
    return 1/(X1-X0)

'''
Construct the Pompeiu derivative
'''

B = 1000
qn = enumq(B)
N = len(qn)
an = a(np.arange(N))
log10rem = log10remainder(N)
print("Max. denominator = %d"%B)
print("Number of terms  = %d"%N)
print("log10(remainder) = %.17g"%log10rem)


X0 = invpompeiu(0,an,qn)
X1 = invpompeiu(1,an,qn)
print("Un-normalized range for g = [%.17g, %.17g]"%(X0,X1))

time1 = time()
print("Calculated normalization in %f s"%(time1-time0))

M = 360360 # LCM of first 15 positive integers
# M = 27720 # LCM of first 12 positive integers
# M = 2520 # LCM of first 10 positive integers
y = np.linspace(0,1,M+1)
print("Plot resolution (full range) = 1/%d (%.17g)"%(M,1/M))

x = normalize(invpompeiu(y,an,qn),X0,X1)
dxdy = dinvpompeiu(y,an,qn) * dnormalize(x,X0,X1)
dydx = 1/dxdy

time2 = time()
print("Calculated window around y E [%.17g, %.17g] (x E [%.17g, %.17g]) in %f s"%(0,1,0,1,time2-time1))

ymin1,ymax1 = 0.45,0.51
y1 = np.linspace(ymin1,ymax1,M+1)
x1 = normalize(invpompeiu(y1,an,qn),X0,X1)
xmin1,xmax1 = normalize(invpompeiu(np.array([ymin1,ymax1]),an,qn),X0,X1)
dxdy1 = dinvpompeiu(y1,an,qn) * dnormalize(x1,X0,X1)
dydx1 = 1/dxdy1

time3 = time()
print("Calculated window around y E [%.17g, %.17g] (x E [%.17g, %.17g]) in %f s"%(ymin1,ymax1,xmin1,xmax1,time3-time2))

ymin2,ymax2 = 0.90,1
y2 = np.linspace(ymin2,ymax2,M+1)
x2 = normalize(invpompeiu(y2,an,qn),X0,X1)
xmin2,xmax2 = normalize(invpompeiu(np.array([ymin2,ymax2]),an,qn),X0,X1)
dxdy2 = dinvpompeiu(y2,an,qn) * dnormalize(x2,X0,X1)
dydx2 = 1/dxdy2

time4 = time()
print("Calculated window around y E [%.17g, %.17g] (x E [%.17g, %.17g]) in %f s"%(ymin2,ymax2,xmin2,xmax2,time4-time3))

'''
Plot
'''

mpl.rcParams['lines.linewidth'] = 0.1
mpl.rcParams['axes.linewidth'] = 0.1
mpl.rcParams['xtick.major.width'] = 0.1
mpl.rcParams['xtick.minor.width'] = 0.1
mpl.rcParams['ytick.major.width'] = 0.1
mpl.rcParams['ytick.minor.width'] = 0.1
mpl.rcParams['axes.xmargin'] = 0
mpl.rcParams['axes.ymargin'] = 0

def set_locator(ax):
    ax.xaxis.set_major_locator(mpl.ticker.MaxNLocator(steps=[1,2,5,10]))
    ax.yaxis.set_major_locator(mpl.ticker.MaxNLocator(steps=[1,2,5,10]))

fig,(ax0,ax1) = plt.subplots(2,1,layout='constrained')
fig.set_size_inches(11,22)

ax0.plot(x,y,linewidth=0.5)
ax0.set_box_aspect(1)
ax0.set_title("$g^{-1}(x)$")
set_locator(ax0)

ax1.plot(x,dydx)
# ax1.set_ymargin(0.05)
ax1.set_ylim(0,2)
ax1.set_box_aspect(1)
ax1.set_title("Derivative")
set_locator(ax1)

ax00 = fig.add_axes([0.115,0.805,0.30,0.15])
ax00.plot(x1,y1,linewidth=0.5)
ax00.set_title("Near $x = %.3g$"%(np.average(ax00.get_xlim())))
ax0.indicate_inset_zoom(ax00,edgecolor='k',alpha=0.2,linewidth=0.5)
set_locator(ax00)

ax10 = fig.add_axes([0.115,0.305,0.30,0.15])
ax10.plot(x1,dydx1)
ax10.set_ylim(0,2.0)
ax10.set_title("Near $x = %.3g$"%(np.average(ax10.get_xlim())))
ax1.indicate_inset_zoom(ax10,edgecolor='k',alpha=0.2,linewidth=0.5)
set_locator(ax10)

ax01 = fig.add_axes([0.585,0.545,0.30,0.15])
ax01.plot(x2,y2,linewidth=0.5)
ax01.set_title("Upper end")
ax0.indicate_inset_zoom(ax01,edgecolor='k',alpha=0.2,linewidth=0.5)
set_locator(ax01)

ax11 = fig.add_axes([0.585,0.305,0.30,0.15])
ax11.plot(x2,dydx2)
ax11.set_ylim(0,2.0)
ax11.set_title("Upper end")
ax1.indicate_inset_zoom(ax11,edgecolor='k',alpha=0.2,linewidth=0.5)
set_locator(ax01)

fig.savefig('pompeiu_derivative.svg')

time5 = time()
print("Generated plots in %f s"%(time5-time4))

Data

Output

Max. denominator = 1000
Number of terms  = 304193
log10(remainder) = -1325.7432497874997
Un-normalized range for g = [-74.622263511656001, 75.732864417864576]
Calculated normalization in 0.049564 s
Plot resolution (full range) = 1/360360 (2.775002775002775e-06)
Calculated window around y E [0, 1] (x E [0, 1]) in 730.076700 s
Calculated window around y E [0.45000000000000001, 0.51000000000000001] (x E [0.4575869955522347, 0.51783956054157898]) in 725.421788 s
Calculated window around y E [0.90000000000000002, 1] (x E [0.9180436035406323, 1]) in 739.555730 s
Generated plots in 0.396963 s

Licensing

I, the copyright holder of this work, hereby publish it under the following license:

	This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication.
	The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission. CC0falsefalse

File history

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	Date/Time	Thumbnail	Dimensions	User	Comment
current	10:21, 28 March 2026		1,056 × 2,112 (589 KB)	wikimediacommons>LaundryPizza03	Increase linewidth of non-derivative plot.

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28 March 2026

603,307 byte

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