Help solving this problem
Let $g:\mathbb{R}\times(0,\infty)\rightarrow \mathbb{R}$ be defined by $g(x,y)=|xy-1|$, $f:\mathbb{R}\rightarrow\mathbb{R}$ by $f(x):=\inf_{y>0}g(x,y)$. Show $f$ is NOT lower semicontinuous.
I have been trying to prove this statement for a couple of days now, I have not gone too far working with it. I am trying to prove it by contradiction but I have not gotten any so far.
I have tried to show that if I assumed $f$ to be lower semicontinuous, then it must be lower semicontinous at $0$, and by that I have tried to get contradiction but I have failed at providing one.
Any help or ideas will be appreciated. I have tried to prove it using different equivalent definitions for lower semincontinuity but as I mentioned, I have not been able to accomplish this task.
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 $x \in U$.