Mathematics
The Weierstrass Function and the Concept of Nowhere Differentiability
Quick fact
In 1872, Karl Weierstrass presented a function that is continuous everywhere but differentiable nowhere, shattering the long-held belief that every continuous function must be differentiable at most points. His function is defined as an infinite sum of cosine waves with carefully chosen amplitudes and frequencies.
Why this is interesting
You've probably been taught that a continuous curve has a tangent line at almost every point. But what if a curve were so jagged that it had no tangent anywhere—not even at a single point?
Read the full explanation
Understanding The Weierstrass Function and the Concept of Nowhere Differentiability
Imagine drawing a smooth curve like a parabola. At every point you can draw a tangent line that just touches the curve. Now imagine a curve with sharp corners, like the absolute value function |x| at 0: there, you can't decide which tangent line to draw because the two sides have different slopes. But the Weierstrass function is far worse—it has corners at every single point. The function is built by adding together an infinite series of cosine waves. Each cosine wave is smooth and wiggly, but when you add infinitely many of them, with the frequency growing much faster than the amplitude, the sum becomes so wiggly that it resembles a jagged mountain range. The key is that the waves are scaled down in size (amplitude) but spaced out in a way that the wiggles become infinitely dense. As you zoom in on any part of the graph, you never see a straight line—no matter how close you look, the curve keeps wiggling with smaller and smaller wiggles. This is what 'nowhere differentiable' means: there is no point where you can define a unique tangent line.
A deeper explanation
The Weierstrass function is defined as W(x) = Σ{n=0}^∞ a^n cos(b^n π x), where 0 < a < 1, b is an odd integer, and ab 1 + 3π/2. The bizarre behavior comes from the tension between two forces. The factor a^n shrinks each term, keeping the total sum finite and making the function continuous. The factor b^n grows the frequency enormously, making the oscillation faster and faster. Because b grows much faster than a decays, the wiggles become so rapid that the graph's slope at any point is undefined: the difference quotients [W(x+h)-W(x)]/h do not converge as h→0, because the high-frequency terms create infinitely many oscillations in any arbitrarily small interval. The continuity follows from the uniform convergence of the series—the partial sums converge uniformly, and the limit of uniformly convergent continuous functions is continuous. But differentiability fails because the sum of derivatives diverges. This counterexample forced mathematicians to stop relying on geometric intuition and to develop rigorous definitions of continuity, differentiability, and convergence. It also paved the way for studying 'pathological' functions and eventually led to fractal geometry.