Is there a Weierstrass function with asymptotic line?

Question

I haven't seen a Weierstrass function with asymptotes. They are all fractals and the sum of trigonometric functions.

If you know a function that is nowhere-differentiable but is continuous and has asymptotes, please let me know, and if not, please explain why.

Answer 1

Let $f : \mathbb (0,1) \to \mathbb R$ be continuous and nowhere differentiable. Then $f_2(x) = \arctan(f(x))$ is continuous, nowhere differentiable, and bounded. Then $$f_3(x) = f_2(x)+\frac{1}{x}$$ is continous, nowhere differentiable, and has vertical asymptote at $x=0$.

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Let $g :\mathbb R \to \mathbb R$ be continuous and nowhere differentiable. Then $g_2(x) = \arctan(g(x))$ is continuous, nowhere differentiable, and bounded. Then $$g_3(x) = e^{-x}g_2(x)$$ is continous, nowhere differentiable, and has horizontal asymptote as $x \to +\infty$.