  We investigate existence, uniqueness and approximation of solutions to
stochastic delay differential equations (SDDEs) under Carath\'eodory-type drift
coefficients. Moreover, we also assume that both drift $f=f(t,x,z)$ and
diffusion $g=g(t,x,z)$ coefficient are Lipschitz continuous with respect to the
space variable $x$, but only H\"older continuous with respect to the delay
variable $z$. We provide a construction of randomized Euler scheme for
approximation of solutions of Carath\'eodory SDDEs, and investigate its upper
error bound. Finally, we report results of numerical experiments that confirm
our theoretical findings.
