Is there a non-linear differentiable real function $f:\mathbb{R}\to\mathbb{R}$ so that every $x \in \mathbb{R}$ has the following property:
in every neighbourhood of $x$, the tangent line to $f$ at $x$ intersects at least one other (and therefore infinitely many) points of the graph of $f$? Or in other words, there is no tangent line of $f$ that is "undisturbed" by other points of $f$ in a neighbourhood centred at where the tangent line meets the curve.
I was thinking about starting by drawing a sine curve and then drawing a courser sine curve that wraps around the previous sine curve and repeating ad infinitum, but I'm not sure this would work, or if the function would remain everywhere differentiable.
Maybe some function to do with the Pompeiu derivative can satisfy the requirements?
Edit: Also vaguely relevant: Differentiable function for which the tangent at each point has infinitely many common points with the graph
Not a complete answer. I solve this question if we assume $f$ is twice differentiable.
No. It suffices to show that $f''(x) = 0$ whenever $f$ has the described property at $x$. Note (1) $f(x+h) = f(x)+hf'(x)+\frac{1}{2}h^2 f''(x)+o(h^2)$ as $h \to 0$; the property means $f(x+h) = f(x)+hf'(x)$ for some arbitrarily small $h$, so we can divide (1) by $h^2$ and take the limit along such $h$ to see that $f''(x) = 0$.