  New explicit exponential stability conditions are presented for the
non-autonomous scalar linear functional differential equation $$ \dot{x}(t)+
\sum_{k=1}^m a_k(t)x(h_k(t))+\int_{g(t)}^t K(t,s) x(s)ds=0, $$ where
$h_k(t)\leq t$, $g(t)\leq t$, $a_k(\cdot)$ and the kernel $K(\cdot,\cdot)$ are
oscillatory and, generally, discontinuous functions. The proofs are based on
establishing boundedness of solutions and later using the exponential dichotomy
for linear equations stating that either the homogeneous equation is
exponentially stable or a non-homogeneous equation has an unbounded solution
for some bounded right-hand side. Explicit tests are applied to models of
population dynamics, such as controlled Hutchinson and Mackey-Glass equations.
The results are illustrated with numerical examples, and connection to known
tests is discussed.
