Weierstrass' function is an example of a function that is continuous, but nowhere differentiable, and can be visualized as being "infinitely wrinkled". I'm having trouble, however, imagining how the integral of such a function would appear. All the techniques that I know of for approximating functions (Taylor series, etc.) would fail on this one. How can this be visualized?
Calculus – What Does the Antiderivative of a Continuous-But-Nowhere-Differentiable Function Look Like?
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Best Answer
I plotted the Weierstrass function $f(x) = \displaystyle\sum_{n = 0}^{\infty}\dfrac{1}{2^n}\cos(3^n\pi x)$ and its antiderivative $F(x) = \displaystyle\sum_{n = 0}^{\infty}\dfrac{1}{6^n\pi}\sin(3^n\pi x)$. Here is what they look like:
The antiderivative of the Weierstrass function is fairly smooth, i.e. not too many sharp changes in slope. This just means that the Weierstrass function doesn't rapidly change values (except in a few places).