This is the definition of inverse continuous function. But, I don't really understand why it defines in this way. For example, What does $s$ mean here? and how does $D_s=\inf\{t:A_t \ge s\}$ make sense? Could you help me intuitively understand this definition?
Thank you in advance!
$s$ is a variable and we are trying define a function $C(s)$ or $C_s$. Since $A_t$ is an increasing function $\{t:A_t>s\}$ is (for any given $s$) a set of the form $[a,\infty)$ or of the from $(a,\infty)$. [Can you verify this? Draw the graph, if necessary]. We define $C(s)$ as the left end-point $a$ of this interval. If $A_t$ is a nice function (say continous and strictly increasing) then this new function would be just the inverse function in the ordinary sense.