  If $f:[a,b]\to \mathbb{R}$, with $a<b$, is continuous and such that $a$ and
$b$ are mapped in opposite directions by $f$, then $f$ has a fixed point in
$I$. Suppose that $f:\mathbb{C}\to\mathbb{C}$ is map and $X$ is a continuum. We
extend the above for certain continuous maps of dendrites $X\to D, X\subset D$
and for positively oriented maps $f:X\to \mathbb{C}, X\subset \mathbb{C}$ with
the continuum $X$ not necessarily invariant. Then we show that in certain cases
a holomorphic map $f:\mathbb{C}\to\mathbb{C}$ must have a fixed point $a$ in a
continuum $X$ so that either $a\in \mathrm{Int}(X)$ or $f$ exhibits rotation at
$a$.
