I am considering a two-varibale function $f:X\times Y\to\mathbb{R}$ where $X,Y$ are two Banach spaces.
Assume that $x\mapsto f(x,y)$ is continuous for all $y\in Y$ and $y\mapsto f(x,y)$ is lower semicontinuous for all $x\in X$. Can I derive that $f(x,y)$ is lower semicontinuous on $X\times Y$?
My attempt is that let $(x_{n},y_{n})$ be a sequence converging to $(x,y)$ in $X\times Y$, so I should have $x_{n}\to x$ in $X$ and $y_{n}\to y$ in $Y$. Now, goal is to prove
\begin{align}
\liminf_{n}f(x_{n},y_{n})\geq f(x,y)
\end{align}
For fixed $\bar x$ and $\bar y$, continuity and lower semicontinuity implies
\begin{align}
\lim_{n} f(x_{n},\bar y)= f(x,\bar y),\liminf_{n} f(\bar x, y_{n})\geq f(\bar x, y)
\end{align}
Combing these two, I should have
\begin{align}
\lim_{i}\liminf_{j}f(x_{i},y_{j})\geq f(x,y), \liminf_{j}\lim_{i}f(x_{i},y_{j})\geq f(x,y).
\end{align}
Then I have no idea whether this can give the final liminf inequality I want or not...
Can somenone help me with this? Or maybe just give a counterexample saying that we can not have the joint lower semicontinuity?
Okay, thanks for the counterexample, so now I know I can not obtain joint lower semicontinuity from continuity + lower semicontinuity. Meanwhile, I have a new question now that what extra condition can we add in order to obtain the joint lower semicontinuity?
