Timeline is a great book, but it firmly belongs in the category of fiction. I believe Michael Crichton was taking a bit of literary license to describe a universe where time travel can occur, whereas there is no current evidence or theory that suggests it is remotely possible. Crichton seems to suggest that time exists completely outside the spatial dimensions, however Einstein's theory of General Relativity, which combines time and space, is strongly supported by experimental observation. Therefore, the basic premise of what Crichton wrote should not be considered scientifically valid, but instead, a clever science fiction twist.
There are many different ways we can look at time, and even today it's a hot topic for modern research.According to Einstein's Theory of Relativity, we can think of time as another dimension, similar to the three spatial dimensions we're familiar with. It should be noted, however, that although time can be looked at as another dimension, it is given special treatment in the Theory of Relativity, and is not on exactly equal footing with spatial dimensions. But from the perspective of relativity, we can say that time is another dimension.
There's another way we can think about time, and that's in regards to something called entropy. Entropy is essentially a measure of the disorder in a system. The more entropy something has, the more disorder that is present (roughly speaking). From everyday experiences like breaking an egg, we see that there's something else special about time. It only flows in one direction. A broken egg will never spontaneously reassemble itself. Time has a preferred direction, which is something physicists call the "Arrow of Time." Although this fact seems obvious in light of the broken egg example, scientists aren't quite sure why time has a preferred direction.
Your last question is in regards to whether time could be considered "part of the continuum." Although we like to think of the spatial dimensions we live in as being a nice, smooth, continuous space, we know from quantum mechanics that on the very small level space can look somewhat discrete, or "blocky." In fact, our usual notions of what a dimension is sometimes have to be abandoned when we're investigating the world on the very small scale. So we see that there isn't even agreement between two of the most successful modern theories of our physical world. This is something that troubles many physicists, and there is a lot of active research that is attempting to discover one theory that describes both the very big (currently described by relativity) and the very small (currently described by quantum mechanics). String theory is one candidate, but whatever theory is able to merge relativity with quantum mechanics will likely shed a lot of insight into the nature of time.
From a scientific perspective, time does exist, and we know this from our own experience. Time has a preferred direction, and it's perfectly fine to think of time as something that is "flowing," in the sense that the current conditions in the world change from moment to moment. However, every indication scientists have discovered so far tell us that time travel (at least traveling back in time to something that already happened) is not possible. This is again something we can see from experience - unfortunately we've yet to meet any time travelers! But although scientists are pretty sure that we cannot travel back in time, there are still many aspects to time we've yet to understand.
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