I've solved this problem, but why is this differentiable?

I've solved this problem, but why is this differentiable?

Let $\alpha:\mathbb R\rightarrow \mathbb R^3$ be a smooth curve (i.e.,
$\alpha \in C^\infty(\mathbb R)$). Suppose there exists $X_0$ such that
for every normal line to $\alpha$, $X_0$ belongs to it. Show that $\alpha$
is part of a circumference.
Part of solution: Let $n(s)$ be the normal vector to $\alpha$ on $s$. For
every $s \in \mathbb R$ there exists an unique real number $\lambda$ such
that $\alpha(s)+\lambda n(s)=X_0$. Now let $\lambda(s)$ be the function
that associates every real number $s$ to this number.



I've finished this question, but I have derivated $\lambda(s)$ and I don't
know why I can do it. Can someone tell me why is $\lambda$ differentiable?