1. **Stating the problem:** We have a cumulative distribution function (CDF) $F:\mathbb{R} \to \mathbb{R}$ and a function $b:[0,1] \to \mathbb{R}$ defined by $$ b(c) = \begin{cases} 0 & \text{if } c=0 \\ \inf F^{-1}([c,1]) & \text{if } c \in (0,1] \end{cases} $$ We want to understand the behavior of $b$ when $F$ has a jump at some point $x$, i.e., when $c=F(x) > a \geq F(x-)$. 2. **Understanding the definitions:** - $F$ is a CDF, so it is non-decreasing and right-continuous. - $F(x-)$ is the left limit of $F$ at $x$. - A jump at $x$ means $F$ increases suddenly from $a$ to $c$ at $x$. - $F^{-1}([c,1])$ is the set of all $x$ such that $F(x) \geq c$. - $b(c)$ is the infimum (greatest lower bound) of this set. 3. **Intuition about $b$:** - $b(c)$ gives the smallest $x$ where $F(x) \geq c$. - Since $F$ is non-decreasing, $b$ is a kind of inverse function but defined via infimum. 4. **Behavior of $b$ at a jump:** - At the jump, $F$ jumps from $a$ to $c$ at $x$. - For $c$ in $(a,c]$, the set $F^{-1}([c,1])$ includes $x$ and points to the right. - For $c$ in $(a,c)$, $b(c)$ is constant and equals $x$ because the infimum of $F^{-1}([c,1])$ is $x$. - At $a$, $b$ jumps because the infimum changes abruptly. 5. **Answering the multiple choice:** - $b$ has a jump at $c$ (true, because $F$ jumps at $x$ and $b$ reflects this). - $b$ has a jump at $a$ (true, since $a$ is the left limit of the jump). - $b$ is strictly increasing over $(a,c)$ (false, it is constant there). - $b$ is constant over $(a,c)$ (true, as explained). **Final conclusion:** $b$ has jumps at $a$ and $c$ and is constant over $(a,c)$. This matches the intuition that $b$ is the generalized inverse of $F$ and reflects jumps in $F$ as jumps and flat parts in $b$.