John Doe
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I'd just like to interject for a moment. What you're referring to as calculus, is in fact, real analysis, or as I've recently taken to calling it, $$\left( \mathbf R,\, +,\, \times,\, \leqslant,\, \left| \cdot \right|,\, \tau \,=\, \left\{ A \,\subset\, \mathbf R \mid \forall x\,\in\, A,\, \exists \varepsilon \,>\, 0,\, \left] x \,-\, \varepsilon,\, x \,+\, \varepsilon\right[ \,\subset\, A \right\},\, \bigcap_{\begin{array}{c} A \,\sigma \text{-algebra of}\, \mathbf R \\ \tau \,\subset\, A \end{array}} A,\, \ell \right)$$-analysis. Calculus is not a branch of mathematics unto itself, but rather another application of a fully functioning analysis made useful by topology, measure theory and vital $$\mathbf R$$-related properties comprising a full number field as defined by pure mathematics.

Many mathematics students and professors use applications of real analysis every day, without realizing it. Through a peculiar turn of events, the application of real analysis which is widely used today is often called "Calculus", and many of its users are not aware that it is merely a part of real analysis, developed by the Nicolas Bourbaki group.

There really is a calculus, and these people are using it, but it is just a part of the field they use. Calculus is the computation process: the set of rules and formulae that allow the mathematical mind to derive numerical formulae from other numerical formulae. The computation process is an essential part of a branch of mathematics, but useless by itself; it can only function in the context of a complete number field. Calculus is normally used in combination with the real number field, its topology and its measured space: the whole system is basically real numbers with analytical methods and properties added, or real analysis. All the so-called calculus problems are really problems of real analysis.

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