Example of a $\mathcal{C}^0$-Caratheodory function

Question

Let $\Omega$ be a open bounded subset of $\mathbb{R}^m, m\in\mathbb{N}$. A function $f:\Omega\times\mathbb{R}^m\to\mathbb{R}$ is a $\mathcal{C}^0$-Caratheodory function if

1. $f(\cdot, \nu):x\in\Omega\mapsto f(x, \nu)\in\mathbb{R}$ is measurable for all $\nu\in\mathbb{R}^m$;
2. $f(x,\cdot):\nu\in\mathbb{R}^m\mapsto f(x, \nu)\in\mathbb{R}$ is $\mathcal{C}^0$ for a.e. $x\in\Omega$.

Could anyone please give me an example of a function which is $\mathcal{C}^0$-Caratheodory and of a function which is not $\mathcal{C}^0$-Caratheodory?

If I understand it, they are not too regular functions, isn't it?

Thank you in advance!

Answer 1

1. Take $g$ continuous, define $f(x,\nu):= g(x)$.

2. Take $h$ non-measurable, define $f(x,\nu):=h(\nu)$.