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Continuous Calculus [pdf]
- schoen 6y agoThis book formulates calculus by starting with the concept of continuity instead of the concept of limits. (It shows that limits can be defined in terms of continuity, rather than the other way away, which other texts usually do.) The simple idea that substitutes for limits in the definition of the derivative is that the derivative of f(x) at x₀ is the value of d that would make (f(x)-f(x₀))/(x-x₀) continuous at x₀ if (f(x-x₀))/(x-x₀) were to equal d (instead of being undefined) when x=x₀. That is to say, there are approximations of the tangent slope on both sides. The derivative, if it exists, is the value of the tangent slope that would make these approximations continuous. The book credits Carathéodory with this concept > that f:ℝ→ℝ has a derivative at a∈ℝ if there exists a continuous function q:ℝ→ℝ such that f(x) = f(a) + q(x)(x−a) and in this case that that derivative is q(a). I actually like this a lot; it feels very elegant.