I'm wondering if there are any points in the Mandelbrot set that are continuous or differentiable? Precisely, is there some point $x_0$ in the set such that there is a continuous/differentiable/smooth function $\varphi: [0,1] \rightarrow M$ such that $\varphi([0,1]) \in M$? I'm not sure I'm phrasing this correctly, but geometrically, if we "zoom in" enough around some point in the Mandelbrot set, do we see the image of a continuous/differentiable/smooth function? Please let me know if this is unclear.
2026-03-25 20:36:38.1774470998
Are there any differentiable points on the boundary of the Mandelbrot set?
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