I think I know such a function for the Mandelbrot set: <a href="http:&#x2F;&#x2F;iquilezles.org&#x2F;www&#x2F;articles&#x2F;distancefractals&#x2F;distancefractals.htm" rel="nofollow">http:&#x2F;&#x2F;iquilezles.org&#x2F;www&#x2F;articles&#x2F;distancefractals&#x2F;distance...</a> (Well, the function that this one converges to.) Not sure about it&#x27;s differentiability, though.