If $f(t,u)$ is continuous wrt. $t$ (and $u$), then is $\sup_{u \in H^1(\Omega)} f(t,u)$ continuous wrt. $t$?
I am unable to prove this. Help appreciated.
If $f(t,u)$ is continuous wrt. $t$ (and $u$), then is $\sup_{u \in H^1(\Omega)} f(t,u)$ continuous wrt. $t$?
I am unable to prove this. Help appreciated.
Let $S= (0,\infty) \times (0,\infty)$. If $f\colon S\to\mathbb R$ is defined by $ f(t,u) = \begin{cases} 0, & \text{if } t\le 1, \\ (t-1)u, & \text{if } t>1 \text{ and } (t-1)u\le 1, \\ 1, & \text{if } t>1 \text{ and } (t-1)u>1, \end{cases} $ then $ \sup_{u\in (0,\infty)} f(t,u) = \begin{cases} 0, & \text{if } t\le 1, \\ 1, & \text{if } t>1 \end{cases} $ is not continuous.