## Algebraic Approach to the Derivative and Continuity

**Authors:** Thomas Colignatus

Continuity is relevant for the real numbers and functions, namely to understand singularities and jumps. The standard approach first defines the notion of a limit and then defines continuity using limits. Surprisingly, Vredenduin (1969), Van der Blij (1970) and Van Dormolen (1970), in main Dutch texts about didactics of mathematics (journal Euclides and Wansink (1970, volume III)), work reversely for highschool students: they assume continuity and define the limit in terms of the notion of continuity. Vredenduin (1969) also prefers to set the value at the limit point (x = a) instead of getting close to it (x → a). Their approach fits the algebraic approach to the derivative, presented since 2007. Conclusions are: (1) The didactic discussions by Vredenduin (1969), Van der Blij (1970) and Van Dormolen (1970) provide support for the algebraic approach to the derivative. (2) For education, it is best and feasible to start with continuity, first for the reals, and then show how this transfers to functions. (3) The notion of a limit can be defined using continuity. The main reason to mention the notion of a limit at all is to link up with the discussion about limits elsewhere (say on the internet). Later, students may see the standard approach. (4) Education has not much use for limits since one will look at continuity. The relevant use of limits is for infinity.

**Comments:** 14 Pages.

**Download:** **PDF**

### Submission history

[v1] 2016-12-02 05:09:11

**Unique-IP document downloads:** 30 times

Vixra.org is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary.
In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution.
Vixra.org will not be responsible for any consequences of actions that result from any form of use of any documents on this website.

**Add your own feedback and questions here:**

*You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.*

*
*