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For $f_c(z)=z^2+c$, the Mandelbrot set is $\mathcal{M}=\{c:\{f_c^n(0){\}}_{n\ge 0}\text{ is bounded}\}$. Thus it records the nonescaping finite critical orbit. The connected Julia set criterion for a polynomial says that the corresponding filled Julia set is connected exactly for these quadratic parameters. Parameters with an attracting fixed point form the main cardioid: its multiplier $\lambda$ gives the fixed point $\lambda /2$ and parameter $c=\lambda /2-{\lambda }^2/4$, with $|\lambda |<1$. Its boundary has a cusp at $c=1/4$. There the critical orbit increases along the real interval to the parabolic fixed point $1/2$, so the cusp belongs to the set. The real section extends from $-2$ to $1/4$; the familiar period-two bulb is attached at $-3/4$.

Assume $J(f)$ is connected. Then the filled Julia set $K(f)$ is connected and full, so its complement $A_{\infty }(f)$ is simply connected. The local Bottcher coordinate therefore extends by the functional equation to a conformal isomorphismIf a finite critical point $c$ lay in $A_{\infty }(f)$, differentiatingat $c$ would giveNeither factor on the right vanishes on the exterior disc under a conformal coordinate, a contradiction. Hence every finite critical point lies in $K(f)$. This is one direction of the connected Julia set criterion for a polynomial.