Let $f$ be a continuous function on $\mathbb{R}_+$ into $\mathbb{R}$. Then for every $n\in\mathbb{N}$ $$\inf\{t\in\mathbb{R}_+\colon f(t)\in [n,\infty)\}=\inf\{t\in\mathbb{R}_+\colon f(t)\in (n,\infty)\}\;.$$ Looking at the interval $[n,\infty)$ which is closed on the left, it seems to me that it is only necessary to assume either the right- or left-continuity of $f$. I do not know which one.
I would appreciate some help in understanding why the above equality holds for our (continuous) $f$. Then I think I would be able to figure out if it's only the right- or left-continuity which is necessary.
Your first assumption appears false. For example, if $f(x)$ takes the value 1 for $x \in [0,1]$, and $x$ for $x \in [1,+\infty)$, then taking $n = 1$, your two infima are $0$ and $1$.