What does a no-where twice differentiable function look like?

Question

I can imagine a no-where once differentiable function being something that just looks like a bunch of noise, but when I think about a function that is everywhere once differentiable but no-where twice differentiable that is the integral of some noisy function, I'm not really sure what it looks like, I don't even know if such a function exists, does anyone have some link or PNG of such a function?

They certainly exist. You can construct one by $x \mapsto \int_0^x f(t) \mathrm{d}t$ where $f$ is continuous and nowhere differentiable. — user780256, Apr 29 '20 at 05:58
That simply mentions some random formula, what I am looking for is an actual graph, an actual depiction of what it looks like. A Weierstrass function is mentioned to be no-where differentiable, so immediately I know that all those animations don't address this question. — AskingRandomQuestions, Apr 29 '20 at 06:00
It matches what a Weierstrass function looks like, being what would commonly be considered a "noisy" function. — AskingRandomQuestions, Apr 29 '20 at 07:03
Im not sure at all if its related but there's this pretty picture of a surface with a well defined tangent but not a well defined curvature https://aperiodical.com/2012/05/torus/ — Calvin Khor, Apr 29 '20 at 07:27

score 3 · Accepted Answer · answered Apr 29 '20 at 07:04

Here is an approximation to a Weierstrass function, with $200$ terms, $b=7$, $a=1.01(1+\frac{3}{2}\pi)/b$, and grid resolution of $0.001$:

And here is an approximation to an antiderivative of this function, with the same resolution, parameters, and number of terms:

Those little bumps will get rougher as we add more terms, but not as quickly as they do in the Weierstrass function.

What does a no-where twice differentiable function look like?

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