In 2001, Csörnyei, O'Neil and Preis proved that the composition of any two Darboux Baire-1 functions $[0,1]\rightarrow [0,1]$ possesses a fixed point, solving a long-standing open problem. In 2004 Szuca proved that this result can be generalized to any $f$ in the class $\Cal J$ of functions $[0,1]\rightarrow [0,1]$ with connected $G_\delta$ graph. As a consequence, he proved that for such functions the Sharkovsky theorem is satisfied.
As the main result of this paper we prove that similarly as for
the continuous maps of the interval, any $f$ in $\Cal J$ has positive
topological entropy if and only if it has a periodic point of period
different from $2^n$, for any $n\in\Bbb N$. To do this we show that
using the Bowen's approach it is possible to define topological entropy
for discontinuous maps of a compact metric space with almost all
standard properties. In particular, the variational principle is true,
and consequently, topological entropy is supported by the set of recurrent
points. We also develop theory of recurrent, ω-limit, and nonwandering
points of functions in $\Cal J$ since, in general, the standard results
from the topological dynamics, are not true: For example, there is a
Darboux Baire-1 function $f$ (hence, $f\in\Cal J$) such that neither the
set of recurrent points nor the set of ω-limit points of $f$ are invariant.