  Let $W$ be a standard Brownian motion with $W_0 = 0$ and let $b: \mathbb{R}_+
\to \mathbb{R}$ be a continuous function with $b(0) > 0$. The first passage
time (from below) is then defined as \begin{align*} \tau := \inf \{ t \geq 0
\vert W_t \geq b(t) \}. \end{align*} It is well-known that the distribution $F$
of $\tau$ satisfies a set of Fredholm equations of the first kind, which is
used, for example, as a starting point for numerical approaches. For this, it
is fundamental that the Fredholm equations have a unique solution. In this
article, we prove this in a general setting using analytical methods.
