16.3 Idle Comments
The concept of limit allows calculus to be discussed without reference to scale. That is, if we are interested in a function \(f(x)\), then we can change \(x\) to \(y\) with \(y=cx\), and the resulting function of \(y\) has the same differentiability properties as \(f\), only with a different derivative.
In the real world, we can use calculus in contexts in which this is not so. Here are some examples. To a scale appropriate to discussing the solar system, the earth has a smooth and differentiable surface, and is roughly spherical. To the scale of us poor mortals, this is quite untrue: there are mountains, tall buildings, holes in the ground, trees, eaves of buildings, and living creatures, and the surface defined by these at any given instant is not differentiable at all. The top of a kitchen table may look flat to us, and its surface is differentiable, but at an atomic scale it is full of holes and whatnot, and at a subatomic scale we have no idea what it looks like. When we store values of a function on a computer, when those values in actuality are irrational, what is stored differs by seemingly random amounts from the actual values, and differences between data points are meaningless at a scale of the size of those differences.
These facts do not take away from the value of the attempt to make calculus rigorous and scale independent. It would be awkward and annoying to us if we had to describe the scale at which functions are continuous or differentiable. It is fairer to say that the functions we define are differentiable at any scale but the models we use to describe reality agree with these functions only at appropriate scales, which phenomena we can look into when we choose to do so.
They also mean that we can use calculus on functions that represent quantities that "look like a straight line" to the scale we are concerned with, even when they do not do so at an infinitesimal scale. It also justifies our trying to draw conclusions from data by staying as far as we possibly can from the limit of small \(d\), something we did with numerical computations.