Your question isn't about smoothness as such, which has various technical definitions in mathematics. But not generally in physics where they think of everything as being smooth enough. As virmior points out, the name of the idea that you're probing is called uniformity - which at least in ordinary language is almost synonymous with smoothness.
The idea of something being uniform is a pervasive idea in mathematics - which one understands by examining the nomenclature of the subject where adjectives such as uniform, standard, normal, smooth abound.
But as you've touched upon smoothness, and though it isn't the main intent of the question it is something that I'm deeply interested in, so I'll tackle that first:
Smoothness has a long intricate history. One might begin with Archimedes method of exhaustion which found the area of a circle by starting with a polygon and increasing the number of sides. This in hindsight is the idea of a limit which was implicit in the foundational work of Newton & Liebniz on the calculus.
In contemporary mainstream language this is now thought of in terms of smooth manifolds, and almost all of physics is written in it implicitly: the smooth spacetime manifold of Einsteins, the smooth bundles of Yang-Mills and so on.
A curve is smooth when you look at a small bit of it it looks almost straight - this is where linearity comes in. The generalisation to higher dimensions is almost obvious: The surface of a sphere is smooth when a small section of it looks almost flat, and so on. This is in fact how mathematicians & physicists think of smooth manifolds.
Until quite recently smoothness was seen as a structure additional to continuity. The Koch snowflake has a fractal boundary which is definitely not smooth. But recent work has shown that they can be conceptualised independently.
Also the category of smooth manifolds doesn't have good categorical properties, for example quotients usually give bad behaviour and pathologies. This has led to revisiting smoothness in terms of universal properties - and results in the quasi-topoi of diffeological manifolds - that is it has almost all the good properties of generalised set theories - aka topoi.
Another way to conceptualise smoothness is to take quite seriously the idea of linearity by positing axiomatically the microline - an infinitesimal line: That is it has no length and yet it is not the point but a line. To make this work one has to drop the law of the excluded middle and adopt intuitionistic logic.
One thinks of here of Deleuzes remark on the differential calculus that he made in difference and Repetition:
it is a mistake to tie the value of the symbol dx to the existence of infinitesimals; but it is also a mistake to refuse it any ontological or gnoseological value...A great deal of heart and a great deal of truly philosophical naivety is needed in order to take the symbol dx seriously...
Linearity & nonlinearity can interact in quite fruitful ways. For example one can have two functions f,g whose behaviour is fractal, but performing the sum f+g is quite-straightforward. This is how linearity enters QM, the wave-functions themselves can have, and will have, pace the simple examples of QM textbooks, exceedingly intricate behaviour.
Returning to Antiquity, one recalls that Aristotle argued against the atomists by arguing for infinite divisibility (this is consistent with his position on the infinite as being potential and not actual). One can say that he is arguing that the line cannot simply be made of points. It must have some kind of cohesion. At this point one can think of topology giving cohesion, or the microline, or Lawveres theory of cohesion in Topoi.
Smoothness generalises metaphysically into the idea of uniformity of nature. This is a background assumption made by scientists, as Virmaoir noted: Not only are manifolds smooth, they are uniform, since any little bit looks like any other little bit. Similarly one thinks that the little patch of space I'm sitting in is similar in all significant respects to some patch near the star Betelgeuse. This is an important distinction that divides Antiquity from today. Aristotle for example divided the universe into the celestial and terrestial, and it was Newton that brought them back together again.
Modelling nature is an essential part of physics. What it means to be a physical law and how accurately or how well it captures nature is a big subject in itself. The SEP carries a page on it. Induction is a crucial part of this, in fact its part of an induction and deduction dialectic. Hume quite famously pointed out that induction can have no logical validity, in the philosophical sense; it is obviously valid as it works - the question is to try to work out why. Hume himself pointed to the psychology of the human mind, and Kant probed further and put certain conceptualisations we must make to make experience even possible below the consciousness of the human mind, in fact below the unconscious, in Freuds taxonomy of the mind. He calls it the intuition.
Kuhn analysed change in scientific knowledge & culture in Marxist terms, calling the crucial event a paradigm change. In this evental history of science the originary event of science that the world is rationally explicable to some degree is associated with the rise of philosophic culture in the lands bordering Mediterrenean.
Its probably worth looking at larger questions connecting the social with the physical. Feyerabend, for example, a philosopher of science connected it to larger questions of politics. One might also note that we call it a law of nature. And laws of course were made by man to be enforced uniformly in a polis.