BodyandSoul vs Standard Calculus 0

BodyandSoul is Constructive or Computational Calculus while Standard Calculus as presented in the standard text book Calculus: A Complete Course by Adams and Essex, can be described as Analytic or Symbolic Calculus. I gave a basic example of the difference between a constructive and symbolic approach in the post

• Mathematics: Backward Magic or Forward Reason

about the Fundamental Theorem of Calculus. Let me here supplement by exhibiting the difference in the approaches as concerns the basic concepts of continuous function and derivate. 

In Analytical/Symbolic Calculus, a real-valued function $s\rightarrow f(s)$ with $s$ a real variable, is said to be continuous at $t$ if

• $f(t) = \lim_{\Delta t\rightarrow 0}f(t+\Delta t)$, 

  and to have derivative $Df(t)$ at $t$ if

• $Df(t) = \lim_{\Delta t\rightarrow 0}\frac{f(t+\Delta t) -f(t)}{\Delta t}$. 

In Constructive/Computational Calculus a real-valued function $s\rightarrow f(s)$ with $s$ a real variable, is said to be Lipschitz continuous with Lipschitz constant $L$ if for all $t$ and $\Delta t$

• $\vert f(t+\Delta t) - f(t)\vert\le L\vert \Delta t\vert$

and differentiable with derivative $Df(t)$ if for a positive constant $K$, for all $t$ and $\Delta t$

• $\vert f(t+\Delta t) - f(t) - Df(t)\Delta t\vert\le K\vert \Delta t^2\vert$.

We see that Analytical/Symbolic Calculus uses the concept of limit which is a difficult concept involving the mysterious process of $\Delta t$ tending to zero or becoming infinitessimally small, but holy God, not zero!

We see that Constructive/Computational Calculus does not use the difficult concept of limit, only the more basic and easy to grasp concept of local change, with a function being Lipschitz continuous if it is locally constant and differentiable if is locally linear with specified deviations.