Weierstrass's Non-Differentiable Function

Speaker: Jeff Calder
Date: December 12, 2014

---

Introduction

• 19th Century Belief: Continuous functions are differentiable at a large set of points.
• Karl Weierstrass (1872): Introduced the first example of a continuous function that is nowhere differentiable.
• Function Definition:
  • ( W(x) = \sum_{n=0}^{\infty} a^n \cos(b^n x) )

Theorem 1 (Weierstrass 1872)

• Conditions:
  • ( a \in (0, 1) )
  • ( b > 1 ) (odd integer)
  • ( ab > 1 + \frac{3}{2} )
• Conclusion: The function W is continuous and nowhere differentiable on ( \mathbb{R} ).

Proof of Continuity

• Convergence:
  • Uses the geometric series ( \sum a^n ) which converges for ( a \in (0, 1) ).
  • The Weierstrass M-test shows uniform convergence to ( W ) on ( \mathbb{R} ).
• Continuity:
  • Each ( a^n \cos(b^n x) ) is continuous; thus, ( W ) is the uniform limit of continuous functions.

Motivation for Condition (1)

• Partial Sums:
  • ( W_n(x) = \sum_{k=0}^{n} a^k \cos(b^k x) )
  • ( W'n(x) = \sum{k=0}^{n} (ab)^k \sin(b^k x) )
  • If ( ab < 1 ), ( W ) is differentiable.
  • Godfrey Hardy (1916) showed ( ab \geq 1 ) is sufficient for nowhere differentiability.

The Weierstrass Function as a Fractal

• Visual Property:
  • Repeating patterns at every scale.
  • Early example of a fractal.

Cosine Properties

• Identity:
  • ( |\cos(x) - \cos(y)| \leq |x - y| )
• Cosine Periodicity:
  • ( \cos(n + x) = \cos(x) ) if n is even.
  • ( \cos(n + x) = -\cos(x) ) if n is odd.

Proof of Theorem 1

• Key Steps:
  1. Bound |A|: Using triangle inequality and identity for cosine.
     • Derive bounds using geometric series.
  2. Bound |B|:
     • Use properties of ( x_m ) and cosine to establish non-negativity.
  3. Combine Bounds:
     • Ensure the difference quotient tends to infinity for ( ab > 1 + \frac{3}{2} ).

Conclusion

• Non-Differentiability:
  • ( \lim_{m \to \infty} \frac{W(x_m) - W(x_0)}{x_m - x_0} = +\infty )
• No Vertical Tangent or Infinite Derivative:
  • ( \lim_{x \to x_0} \frac{W(x) - W(x_0)}{x - x_0} ) does not exist.

---

Visual Representation:

• Figure 1: Plots of the Weierstrass function for ( a = 0.5 ) and ( b = 3 ), illustrating self-similarity across scales.