
Continuous Calculus [pdf] - happy-go-lucky
http://www-users.math.umn.edu/~olver/ln_/cc.pdf
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schoen
This 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.

