Problem 3 chapter 12 Jost Postmodern analysis
Solution 1:
We have $g(0,y) = 1$, thus $f(0) = 1$. Moreover, for $x > 0$ we have $g(x,1/x) = 0$, thus $f(x) = 0$. This shows that $f$ is not lower semicontinuous: There is no neighborhood $U$ of $0$ in $\mathbb R$ such that for all $f(x) > f(0) - 1/2$ for all $x \in U$.