We don't know time is continuous at small scales. But I don't think anyone claims it is.
You have asked several questions. But Is time real? does not seem to be one of them
(1) Is time continuous at small scales?
Experience tells us time seems to be continuous at large scales, but that might just be a consequence of it being divided into very small frames too small to measure with current instruments.
For example matter seems continuous at large scales -- like you can keep chopping a block of cheese in half forever -- but on closer inspection the cheese is not continuous. It is broken into atoms and there is a smallest possible amount of cheese you can have.
General relativity relies on time being continuous (at large scales) and gives accurate predictions. This however says nothing about the small scale physics where GR stops being useful. In my experience most mathematicians and scientist only see the continuousness as a useful mathematical tool and not a 100% accurate desciption of reality.
At small scales the properties of space behave strangely. The uncertainty principle from Quantum Mechanics tells us that the exact position of an object has no physical meaning. There are just degrees of certainty. Another problem is it sometimes becomes meaningless to say which of two atoms is which.
So it would not surprise me if the small scale structure of time also behaves strangely. Maybe not discrete, but probably not continuous. Say once you zoom in close enough it becomes meaningless to give the exact moment an event happened.
(2) If time is continuous can we prove it?
Probably not. Any experimental setup will only have finite precision. One might verify time does not behave as though it is broken into 0.0000001 nanosecond chunks, simply by designing a camera that can take that many frames per second, and filming a light beam as it travels across the room. But then what about 0.0000000001 nanosecond chunks?
Suppose there are Doctor Who aliens who, once per minute, appear all over the planet. They give everyone on the planet strange looks and then vanish one micro-nano-attosecond later. With the correct instruments you could prove whether these aliens are real or not. But that says nothing about the aliens that only last one micro-nano-atto-PICOsecond.
Of course this raises the cosmic speed limit problem. Nothing travels faster than the speed of light c and location stops being meaningful at the so-called planck length p. So a universe with frames smaller than c/p might have no physical meaning, since that's too short for anything meaningful to happen.
In this case you ask a question and, rather than have an answer for you to discover, the universe sticks out it's tongue and blows you a big raspberry.
There is also the foundational issue of what you should compare to in order to establish a temporally continuous universe. We set up the experiment such that if the universe is temporally continuous we get outcome X and otherwise we get outcome Y. It's unclear (and perhaps meaningless) how you would predict the behaviour of an experiment in a universe that by definition does not exist.
(3) Is time coherent?
I am not sure what this means. But I will point out that even if there was discrete time, it does not rule out coherence in the sense that each moment influences the next. It's just the influence looks different. For example video game time is discrete but the time is still coherent, in the sense that inputs from the player at one moment influence the world at subsequent moments.