We consider the set valued functions $C,NW,CR$ and
$\Omega $ taking $f$ in $C(I,I)$ to its center $C(f)$, its set of
nonwandering points $NW(f)$, its set of chain recurrent points $CR(f)$
and its collection of $\omega $-limit sets $\Omega (f)=\{\omega
(x,f):x\in I\}$, and consider how these sets are affected by
pertubations of $f$. Our main results characterize those functions $f$
in $C(I,I)$ at which $C$, $NW$ and $\Omega $ are continuous. We also
characterize those functions at which $CR$ are continuous when we
restrict our attention to those functions $f$ in $C(I,I)$ with zero
topological entropy.
Mathematics Subject Classification. Primary 26A18, 54H20, 37B20.