A map $f:X\rightarrow X$ on a topological space $X$ is
called transitive if for any pair of non-empty open sets
$U,V \subset X$, there is $n\in\mathbb{N}$ such that $f^n(U)\cap
V\neq \emptyset$. $f$ admits dense orbit if there is $x\in
X$ such that the orbit of $x$, $Orb(x):=\{ x,f(x),f^2(x),\dots
\}$, is dense in $X$. Given a Baire separable metric space $M$, we
show that any transitive map $f:M\rightarrow M$ with at most one
point of discontinuity, admits points with dense orbit. We also
present an example of a transitive map $f:[0,1]\rightarrow [0,1]$
with two points of discontinuity which admits no dense orbit. This
shows that our result is sharp.