My question is about the function defined in Rudin 4.31, mentioned by this question:
Remark 4.31 in Baby Rudin: How to verify these points?
The function is defined as $$f(x) \colon= \sum_{x_n < x} c_n \ \ \ \ \text{ for all } x \in (a,b).$$
I'm having trouble trying to visualise what such a function would look like in terms of its "smoothness". In particular, I'm interested in the case where $$a = 0$$ $$b = 1$$ $$\{x_n\} = \{\mathrm{rationals\:in\:(0,1)}\}$$ $$c_n = \frac{1}{2^n}$$
My question is
What would the graph of $f$ look like in terms of its "smoothness"?
I thought it would look something like the Weierstrass function but monotonic. It seems like there would have to be some sort of recursive/fractal structure, and yet there also has to be a definite "jump" at each rational point in the domain.
I would appreciate a diagram with an intuitive explanation.
Edit: This image is how I'm trying to imagine it. The blue line is $f(x)$. The total length of the green gaps converge to $2$. The red gaps can't really exist, because the rationals are dense, so I'm wondering if the actual graph when you get rid of the red gaps would look "smooth" or "jagged" (like the Weierstrass function).