newtonian mechanics - Conservation of Linear Momentum at the point of collision

This is a pretty basic conceptual question about the conservation of linear momentum.

Consider an isolated system of 2 fixed-mass particles of masses $m_1$ and $m_2$ moving toward each other with velocities $v_1(t)$ and $v_2(t)$ respectively.

Now conservation of momentum says that at any point during the particles' motion the quantity $$m_1v_1(t) + m_2v_2(t) =constant$$

With non-zero velocities and non-zero masses this constant will be non-zero.

Let us say the particles collide at time $t_0$. At the point of collision, both particles have velocity zero. which would mean that the constant above will be zero. Contradiction.

I realize I might be going wrong in my reasoning at the point of collision.

In fact, I feel defining velocity at that point would not even make sense, since if one considers the displacement functions $x_i(t)$ $i=1,2$ of the particles, then $t_0$ would represent a point of non-differentiability of $x_i(t)$ for $i=1,2$.

So assuming there are no collisions, by following the text-book derivation I can see why

$$m_1v_1(t) + m_2v_2(t) = C_1$$ before the collision and $$m_1v_1(t) + m_2v_2(t) = C_2$$ after the collision

would hold true, but not why $C_1=C_2$

Can someone help me in clearing this up?