Let $D\subset\mathbb{R}^n$ be a compact set of non-zero, $n$-dimensional Lebesgue measure, where $n\ge 2$. Let $f : D\times\mathbb{R}\to\mathbb{R}$ be a Caratheodory function, and let $v:D\to\mathbb{R}$ be continuous on $D$. Is it possible for $f(\cdot, v(\cdot))$ to not be essentially bounded on $D$ ? If so, what is an example of such an unbounded function ?
Note that this question is important to research because such functions arise in many contexts, such as partial differential equations.