This is an old problem. Zeno of Elea is credited with some classical pointed formulations of it about 2500 years ago. (Note: not "Xeno" as one commenter above spelled him).
There are infinitely many different places to be at between 0 cm and 1 cm, but by the same token there are also infinitely many different instants in, say, one second, so they match up nicely.
Now, whether space and time can physically be subdivided infinitely finely is not a mathematical question. It's just that the most common mathematical model of them allows arbitrarily fine divisions, because that is much easier to deal with than the alternative and consistently seems to lead to useful results in practice. It is perfectly conceivable that actual time or space cannot be divided indefinitely; that would just mean that the mathematical model is not an accurate description at small enough scales. (Again, this would not be a mathematical problem. The model might describe something else, or describe no physical situation, and it would be no worse as mathematics for that).
As a physical question, the last few hundred year's physics has shown that matter cannot be subdivided indefinitely; a a scale of around 0.00000001 cm you find atoms that cannot be cut up without fundamentally changing what they are. However, the atoms are still thought to move around in a fundamentally continuous space. That might change with the next unpredictable revolution in physics, of course.