Let
X
X
and
Y
Y
be topological spaces, let
S
S
be a dense subspace of
X
X
, and let
f
:
S
→
Y
f:S \to Y
be continuous. When
Y
Y
is the real line
R
{\mathbf {R}}
, the Lebesgue sets of
f
f
are used to provide necessary and sufficient conditions in order that the (bounded) function
f
f
have a continuous extension over
X
X
. These conditions yield the theorem of Taǐmanov (resp. of Engelking and of Blefko and Mrówka) which characterizes extendibility of
f
f
for
Y
Y
compact (resp. realcompact). In addition, an extension theorem of Blefko and Mrówka is sharpened for the case in which
X
X
is first countable and
Y
Y
is a closed subspace of
R
{\mathbf {R}}
.