  It is shown that for a function $f:\mathbb R^2\to \mathbb R$ which is
measurable with respect to the first variable and upper semicontinuous
quasicontinuous and increasing with respect to the second variable there exists
a Caratheodory's solution $y(x)=y_0+\int\limits_{x_0}^xf(t,y(t))d\mu(t)$ of the
Cauchy problem $y'(x)=f(x,y(x))$ with the initial condition $y(x_0)=y_0$. There
are constructed examples which indicate to essentiality of condition of
increasing and give the negative answer to a question of Z.~Grande.
