The Weierstrass function: continuous, nowhere differentiable

Plots the Weierstrass function sum a^n cos(b^n pi x) at several zoom levels. The graph looks equally rough at every scale, difference quotients grow without bound as the step shrinks, and the graph’s measured length keeps growing as it is sampled more finely.

import matplotlib.pyplot as plt
import numpy as np

from mathematicskit.fractals_chaos import weierstrass_function

Zooming in never smooths the graph

fig, axes = plt.subplots(1, 3, figsize=(12, 3.5))
for ax, width in zip(axes, (2.0, 0.2, 0.02)):
    x = np.linspace(0.3 - width / 2, 0.3 + width / 2, 4000)
    ax.plot(x, weierstrass_function(x), lw=0.7)
    ax.set_title(f"window width {width}")
fig.suptitle("W(x) with a = 0.5, b = 7")

Text(0.5, 0.98, 'W(x) with a = 0.5, b = 7')

Difference quotients diverge

for h in (1e-1, 1e-2, 1e-3, 1e-4, 1e-5):
    slope = (weierstrass_function(0.3 + h) - weierstrass_function(0.3)) / h
    print(f"h = {h:.0e}: difference quotient {float(slope):12.2f}")

h = 1e-01: difference quotient        -9.78
h = 1e-02: difference quotient         5.53
h = 1e-03: difference quotient      -119.59
h = 1e-04: difference quotient      -504.59
h = 1e-05: difference quotient      1190.24

The graph has infinite length

for n in (10**3, 10**4, 10**5, 10**6):
    x = np.linspace(0, 1, n)
    y = weierstrass_function(x)
    length = np.sum(np.hypot(np.diff(x), np.diff(y)))
    print(f"{n:>8d} samples: polygonal length of the graph over [0, 1] = {length:8.1f}")

   1000 samples: polygonal length of the graph over [0, 1] =     96.8
  10000 samples: polygonal length of the graph over [0, 1] =    343.7
 100000 samples: polygonal length of the graph over [0, 1] =   2101.0
1000000 samples: polygonal length of the graph over [0, 1] =  10109.4