What real numbers are in the Mandelbrot set? The Mandelbrot set is defined over the complex numbers and is quite complicated. It's defined by the complex numbers $c$ that remain bounded under the recursion:
$$ z_{n+1} = z_n^2 + c,$$
where $z_1 = 0$.
If $c$ is real, then above recursion will remain real. So for what values of $c$ does the recursion remain bounded?

The Wikipedia page gives the intersection of the set with the real axis as $[-2,0.25]$
Added:  You can verify that $-2$ is in the set easily, and that any more negative number decreases each iteration without bound.  For the positive end, each iteration is greater than the one before.  To hit a limit, you must have $z=z^2+c$, which has the solution $z=\frac{1+\sqrt{1-4c}}2$, which becomes imaginary at $c \gt \frac 14$
