Caratheodory's solution of the Cauchy problem and question Z.Grande

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 $y(x)=y_0+{\int }_{x_0}^{x}f(t,y(t))d\mu (t)$ of the Cauchy problem $y^{\mathrm{\prime }}(x)=f(x,y(x))$ 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.