I am trying to understand the derivation of the CDF of a Brownian Motion absorbed at a value as shown in this link. The derivation in this link claims that for $a>0$, the Brownian Motion absorbed at a value $a$, defined as
$$B_a=\begin{cases}B_t&\text{for }t<T_a,\\ a&\text{for }t\geq T_a,\end{cases}$$
where $T_a=\inf\{t\geq0:B_t=a\}$ (so the event $\{t\geq T_a\}\equiv\{\max_{s\in[0,t]}B_t\geq a\}$), we obtain
$$\mathbb{P}(B_t\geq x)=\Phi\left(\frac{x}{\sqrt{t}}\right)+\Phi\left(\frac{2a-x}{\sqrt{t}}\right)-1.$$
I plotted the CDF, and it's not monotonically increasing. (I also worked out the PDF and it definitely goes negative at some point.) Is there a typo or mistake in the derivation outlined here? I've tried to nitpick at the use of the reflection principle but it seems logically sound to me.
The CDF is $P(B_a(t) \le x)$. The formula for the CDF is correct, and monotone increasing, for $x<a$. The CDF has a jump discontinuity at $x=a$ and it equals 1 for all $x \ge a$.