What is the graph of Rudin's 7.18 function

Question

Code borrowed from here

Theorem 7.18 from Baby Rudin:

There exists a real continuous function on the real line which is nowhere differentiable.

proof

Define $$\tag{34} \varphi(x) = \lvert x \rvert \qquad \qquad (-1 \leq x \leq 1) $$ and extend the definition of $\varphi(x)$ to all real $x$ by requiring that $$ \tag{35} \varphi(x+2) = \varphi(x). $$ Then, for all $s$ and $t$, $$\tag{36} \lvert \varphi(s) - \varphi(t) \rvert \leq \lvert s-t \rvert. $$ In particular, $\varphi$ is continuous on $\mathbb{R}^1$. Define $$ \tag{37} f(x) = \sum_{n=0}^\infty \left( \frac{3}{4} \right)^n \varphi \left( 4^n x \right). $$ Since $0 \leq \varphi \leq 1$, Theorem 7.10 shows that the series (37) converges uniformly on $\mathbb{R}^1$. By Theorem 7.12, $f$ is continuous on $\mathbb{R}^1$. Now fix a real number $x$ and a positive integer $m$. Put $$ \tag{38} \delta_m = \pm \frac{1}{2} \cdot 4^{-m} $$ where the sign is so chosen that no integer lies between $4^m x$ and $4^m \left( x + \delta_m \right)$. This can be done, since $4^m \left\lvert \delta_m \right\rvert = \frac{1}{2}$. Define $$ \tag{39} \gamma_n = \frac{ \varphi \left( 4^n \left( x + \delta_m \right) \right) - \varphi \left( 4^n x \right) }{ \delta_m }. $$ When $n > m$, then $4^n \delta_m$ is an even integer, so that $\gamma_n = 0$. When $0 \leq n \leq m$, (36) implies that $\left\lvert \gamma_n \right\rvert \leq 4^n$. Since $\left\lvert \gamma_m \right\rvert = 4^m$, we conclude that $$ \begin{align} \left\lvert \frac{ f \left( x + \delta_m \right) - f(x) }{ \delta_m } \right\rvert &= \left\lvert \sum_{n=0}^m \left( \frac{3}{4} \right)^n \gamma_n \right\rvert \\ &\geq 3^m - \sum_{n=0}^{m-1} 3^n \\ &= \frac{1}{2} \left( 3^m + 1 \right). \end{align} $$ As $m \to \infty$, $\gamma_m \to 0$. It follows that $f$ is not differentiable at $x$.

I want to see the graph of $f(x)$ I don't have the tools that graph such functions

Answer 1

This is the function after summing the first 25 terms in the series.

And this is the function as we sum from 1 to 15 terms.

Here is the code to generate it:

import numpy as np
import matplotlib.pyplot as plt

from matplotlib.animation import FuncAnimation

def phi(x):
    return np.abs(x % 2 - 1)

def f(x, num_terms=25):
    n = np.arange(num_terms)[:, np.newaxis]
    terms = (3/4)**n * phi(4**n * x)
    return np.sum(terms, axis=0)

# Generate x values
x = np.linspace(-2, 2, 100000)

# Create a figure and axis
fig, ax = plt.subplots(figsize=(8, 6))

# Initialize the plot
line, = ax.plot([], [])

# Set up the plot properties
ax.set_xlim(-2, 2)
ax.set_ylim(0, 5)
ax.set_xlabel('x')
ax.set_ylabel('f(x)')
ax.grid(True)

# Update function for the animation
def update(i):
    y = f(x, i+1)
    line.set_data(x, y)
    ax.set_title(f'Visualization of f(x) with {i+1} terms')
    return line,

# Create the animation
anim = FuncAnimation(fig, update, frames=15, interval=500, blit=True)

# Save the animation as a GIF
anim.save('f(x)_animation.gif', writer='pillow')

# Show the plot
plt.show()