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    $\begingroup$ Yes. A standard theorem (you can find it in most textbooks on approximation theory) is that for every positive continuous function $\delta$ on $\mathbb R$ and any continuous $f$ there, one can find a real analytic (even entire) $g$ such that $|g-f|\le\delta$ on the line. This doesn't yet give you what you want if used just as a black box, but if you look at the proof, you'll be able to modify it pretty easily to answer your question. To get 1-1, it helps to approximate the derivative together with $f$ itself. If you still have trouble with it, let me know and I'll post more details :-) $\endgroup$ Commented Jan 3, 2025 at 2:00
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    $\begingroup$ This is certainly so if the curve is parametrized by a non-compact interval. But if the curve is parametrized by $[a,b]$ the answer is negative. $\endgroup$ Commented Jan 3, 2025 at 15:38
  • $\begingroup$ @fedja Can you say a bit more about how controlling/approximating the derivative allows you to prevent self-intersections between distant input points? $\endgroup$ Commented Jan 3, 2025 at 22:10
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    $\begingroup$ @KevinCasto Distant points are controlled by the small error of the approximation to a continuous curve you draw adding piece by piece already. The derivative is needed only to avoid self-intersections in a very near vicinity (if you have a non-self intersecting continuous $f$, its uniform approximation $g$ certainly won't glue faraway parameters, but can glue very close ones as much as it wants unless you control something like the derivative as well). $\endgroup$ Commented Jan 3, 2025 at 23:23