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"[T]he idea of function as a machine is such a powerful and intuitive one that it tends to be used pretty universally until you have a good reason to abandon it. Non-mathematicians rarely encounter such reasons, even in the more mathematically demanding disciplines like physics, computer science, and engineering. In fact, most of the time we tend to double down and promiscuously apply the "function as machine" picture to operators. If a function is a machine that turns numbers into other numbers, and operator is a machine that turns functions into other functions. One such operator is called the Laplace transform, after the french mathematician Pierre-Simon Laplace. But I think we'll stick to calling these posts Sunday Functions, even if we take the occasional look at operators."
More like this
Let's do two functions today. As sometimes happens, in this case we're not so interested in the functions themselves as the fact that these functions happen to be part of a general class of functions.
I first met this function sometime in the year 2001 in the manual for a graphing calculator. The manual said that the function had no "closed-form analytic antiderivative" but nonetheless the calculator could integrate it numerically.
Since the posts of sheaves have been more than a bit confusing, I'm going to take
the time to go through a couple of examples of real sheaves that are used in
algebraic topology and related fields. Todays example will be the most canonical one:
Draw the graph of a function. Roughly speaking, if there's no holes, jumps, or other choppy weirdness it's a continuous function. The function is connected to itself like a curvy rope laid out on the ground, with no cuts.