Final answer:
Open strings can split at any point and join at the ends when they are close. The likelihood of the two ends of a string meeting at the exact same point is small, but they can be attracted to each other. The rest mass of particles in string theory is determined by the vibrations of the string, not the length.
Step-by-step explanation:
In string theory, open strings can both split at any point and join at the ends when the ends are close to each other. The likelihood of the two ends of a string ending up at the exact same spatial position is indeed very small, but they do not need to meet at the same point. When the ends are close, they can be attracted to each other and join to form a closed string.
When an open string breaks at an arbitrary point, it does not create particles of arbitrary mass. The rest mass of a particle is not directly related to the length of a string. Instead, the different string vibrations determine the properties of the particles, including their masses. Particle masses do not form a continuum because the vibrations of the string are quantized, meaning they can only take on certain discrete values.
It's important to note that string theory is a complex and still untested theory, and many details are still being explored and understood by physicists. The mathematics of string theory is elegant, but experimental evidence is needed to confirm its predictions.