# Disconnected image of a derivative

I would like to know if there exists a differentiable vector-valued function $$f:[a,b]\rightarrow \Bbb R^2$$ such that the image $$f'([a,b])$$ of its derivative is disconnected.

My Attempt

First of all I asked for $$f$$ to be vector-valued because, if it were real-valued, then Darboux's Theorem would imply that $$f'([a,b])$$ be connected. For a similar reason, if we let $$x(t)$$ and $$y(t)$$ denote the component functions of $$f$$, then their derivatives can not both be continuous, for otherwise $$f'$$ would also be continuous hence preserving connectedness. Therefore, WLOG, $$x'$$ has to be discontinuous; however, Darboux's Theorem imply that it can not have discontinuities of the I kind or discontinuities of the II kind with infinite directional limits on $$]a, b[$$. Now, all the functions I came up for $$x$$ are not enough to construct the desired $$f$$ and I am starting to believe that such a function doesn't actually exist but I am not able to prove it.

Any help or hint, as always, is highly appreciated!

• @Aname I do not understand the point you are trying to make Apr 21, 2022 at 10:17

Consider the curve \begin{align*} f(t) &= (t^{2}\sin(1/t), t^{2} \cos(1/t)) \\ &= t^{2}(\sin(1/t), \cos(1/t)), \end{align*} extended to be $$(0,0)$$ at $$0$$. The standard calculation shows \begin{align*} f'(t) &= (2t\sin(1/t) - \cos(1/t), 2t\cos(1/t) + \sin(1/t) \\ &= 2t(\sin(1/t), \cos(1/t)) + (-\cos(1/t), \sin(1/t)) \end{align*} for $$t \neq 0$$ and $$f'(0) = (0, 0)$$.
Since $$|f'(t)| \geq |1 - 2t|$$ by the reverse triangle inequality, in a sufficiently small deleted neighborhood of $$t = 0$$ the derivative is bounded away from $$(0, 0)$$. The image of the derivative is therefore the union of the origin and a (very fast spiral) curve lying in an annulus about the unit circle.
• Thank you very much for the answer! I did think about using the unit circle parametrization by sine and cosine and the function $t^2\sin(1/t)$ but it didn’t came to my mind to combine these two ideas! Apr 21, 2022 at 14:54