At least, that is what most physicists say. But philosophers disagree, usually with deep conviction: "Without continuous time, we cannot think! Without continuous time, nothing makes sense. Time must be continuous - there is no other option!" And many physicists agree.
It is like the old saying: a man cannot live happily with a woman; and a man cannot live happily without one.
You can find both sides discussing fiercely all over the literature (both about time and about women). And all arguments are usually convincing. Therefore, in this discussion, both sides must be right - and both must be wrong. But how?
Reconciling the two sides is the main problem of quantum gravity. It must be done carefully.
- Without continuous time, classical physics, quantum theory and general relativity make no sense. Energy conservation makes no sense. Symmetries make no sense.
- With continuous time, there is no way to resolve the contradictions between quantum theory and general relativity. General relativity does not allow point masses. Short times and distances blow up quantum field theory.
Now come the crazy men. They have proposals en masse:
- Supersymmetry says: space-time has fermionic coordinates (no joke).
- String theorists say: space-time is continuous but 10 or 11 dimensional and fermionic (no joke).
- Non-commutative people say: space-time is non-commutative (no joke).
- Relativity researchers say: space-time has really only 1+1 dimensions (no joke).
So modern fundamental physics has become a world in which daydreamers are normal; they are respected, and worse, they are taken as examples to follow.
Try this out: tell one of these men that his ideas cannot be tested. He will jump at you. Try it. An enraged lion is easier to handle.