Proof that the generalized inverse of an increasing right-continuous function is also right-continuous Let $a:[0,\infty) \to [0,\infty]$ be an increasing right continuous function, and $\tau:[0,\infty) \to [0,\infty]$ be the generalized inverse, i.e. $\tau(s):=\inf\{t\ge 0: a(t)>s\}, \inf \emptyset = \infty$. Prove that $\tau$ is right continuous. 
Proof. $$\{t:a(t)>s\} = \inf_{\epsilon>0} \{t:a(t)>s+\epsilon\}.$$ Therefore, $\inf\{t\ge 0: a(t)>s\} = \inf_{\epsilon>0} \inf \{t\ge 0: a(t)>s+\epsilon\}$ proving right continuity. 
I have trouble figuring out why this last equality proves right-continuity. Is there a way to see this immediately? I would greatly appreciate any help.
 A: Substituting in the definition of $\tau(s)$, that equation is $$\tau(s)=\inf_{\epsilon>0}\tau(s+\epsilon).$$  This means for any $\delta>0$, there exists $\epsilon>0$ such that $\tau(s+\epsilon)<\tau(s)+\delta$.  Since $\tau$ is increasing, this means $\tau(s)\leq\tau(t)<\tau(s)+\delta$ for any $t$ such that $s\leq t<s+\epsilon$.  This exactly verifies the $\epsilon$-$\delta$ definition of right continuity.
A: Proof that $\tau$ is nondecreasing. If $s'\ge s$, then the set inclusion
$\{t:a(t)>s'\}\subset\{t:a(t)>s\}$ implies
$$\tau(s'):=\inf\{t:a(t)>s'\}\ge\inf\{t:a(t)>s\}=:\tau(s).\tag1$$

Proof that $\tau$ is right continuous. For given $s$, suppose $t$ is such that $a(t)>s$. Then there exists $\delta>0$ such that $a(t)>s+\delta$. So
$$t\ge \inf\{t:a(t)>s+\delta\}=:\tau(s+\delta)\ge\inf_{\delta >0}\tau(s+\delta),\tag2$$
so the RHS of (2) is a lower bound for $\{t: a(t)>s\}$. Therefore
$$\inf_{\delta>0}\tau(s+\delta)\le \inf\{t: a(t)>s\}=:\tau(s).\tag3$$
Let $\epsilon>0$. Inequality (3) says there exists $\delta>0$ such that $\tau(s+\delta)<\tau(s)+\epsilon$. If $s\le s'<s+\delta$ then
$$\tau(s)\stackrel{(1)}\le\tau(s')\stackrel{(1)}\le\tau(s+\delta)<\tau(s)+\epsilon;$$ this means that $\tau$ is right continuous at $s$.

Note that these properties follow from the condition $a(t)>s$ and the definition of infimum. Nowhere do we use the assumption that $a$ is increasing or right-continuous.
