Let $\overline{\Omega}\subset\mathbb{R}^{N}$ be a compact set and $T<\infty$. Assume $f : \overline{\Omega}\times(0,T)\to\mathbb{R}$ is continuous and sastifies
(A) $\forall x\in\overline{\Omega}, \sup\limits_{t\in(0,T)}|f(x,t)|<\infty$
(B) $\forall t\in(0,T), \sup\limits_{x\in\overline{\Omega}}|f(x,t)|<\infty$
Is it possible to show that a function defined as $\bar{f}:=\sup\limits_{t\in(0,T)}f(\,\cdot\,,t)$ to be a continuous function in $\overline{\Omega}$? I would like to apply this problem for my PDE research so any help (or counterexample) will be useful. (Remark : I know that $\sup\limits_{x\in\overline{\Omega}}|f(x,\,\cdot\,)|$ is a continuous function in $(0,T)$ since $\overline{\Omega}$ is compact.)
Thank you very much!
No, consider the following counterexample.
Let $f:(0,1)\times [0,1]\longrightarrow \mathbb{R} $ be a function defined as $f(x,y)=\text{arctan}(y/x)$. Then, $\bar{f}$ is not continous on $0$.