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Non-differentiable Continuous Functions

A function can still be continuous and not be differentable at a point. For example, the absolute value function |x| is continuous at x = 0, but the deriviatve is not defined at that point. Here we give an example of a function, the Weierstrass function W, which is continuous everywhere but differentiable nowhere.

The Weierstrass function W defined as by

where 0 < H < 1 and ω > 1 is continuous and nondifferentiable for all values of x. This function is well defined, for example, to 100 digits, we may calculate

However, Figure 1 shows a zoom on the Weierstrass function W0.3, 1.3(x) starting from the interval [0, 1] and slowly zooming by over six orders of magnitude. The red point is (0.5, W0.3, 1.3(0.5)).

Figure 1. A zoom on the point (0.5, W0.3, 1.3(0.5)).

Compare the zoom on the Weierstrass function with the zoom demonstrated on the differentiable functions page.