  A differentiable function is pseudoconvex if and only if its restrictions
over straight lines are pseudoconvex. A differentiable function depending on
one variable, defined on some closed interval $[a,b]$ is pseudoconvex if and
only if there exist some numbers $\alpha$ and $\beta$ such that
$a\le\alpha\le\beta\le b$ and the function is strictly monotone decreasing on
$[a,\alpha]$, it is constant on $[\alpha,\beta]$, the function is strictly
monotone increasing on $[\beta,b]$ and there is no stationary points outside
$[\alpha,\beta]$. This property is very simple. In this paper, we show that a
similar result holds for lower semicontinuous functions, which are pseudoconvex
with respect to the lower Dini derivative. We prove that a function, defined on
some interval, is pseudoconvex if and only if its domain can be split into
three parts such that the function is strictly monotone decreasing in the first
part, constant in the second one, strictly monotone increasing in the third
part, and every stationary point is a global minimizer. Each one or two of
these parts may be empty or degenerate into a single point. Some applications
are derived.
