Timeline for Is simplifying a rational function considered as a continuous extension?
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| Dec 30, 2015 at 15:39 | comment | added | Philipp Imhof | @DanielHast: Most mathematical concepts are too complicated to be taught in all details from the very beginning. Instead, we must simplify them to a certain extent. Or to put it in Bruner's words: «fifth-grade children very rapidly grasped central ideas from the theory of functions, although had the teacher attempted to explain to them what the theory of functions was, he would have drawn a blank. Later [...], the time would be ripe for introducing them to the necessary formalism.» Functions have several aspects and some of them must be postponed. They can easily be learned later. | |
| Dec 30, 2015 at 3:37 | comment | added | Daniel Hast | @DanielR.Collins: I recognize that the intent was to ask about the function defined by that formula according to the aforementioned convention... but the reason there was a question in the first place is because of imprecision created by that convention. | |
| Dec 30, 2015 at 3:31 | comment | added | Daniel R. Collins | So I wonder if @DanielHast also thinks that the OP's $f(x)$ is a formula, and not a function, for like reasons? | |
| Dec 30, 2015 at 3:28 | comment | added | Daniel R. Collins | Agreeing with @mweiss, this is also standard in college calculus texts, e.g.: Stein and Barvellos, Calculus and Analytic Geometry, 5E (1992) Sec 2.1: "The domain consists of all $x$ such that $f(x)$ is defined. So exclude those $x$ for which the formula makes no sense (requiring, for instance, division by 0 or the square root of a negative number)." It's pretty reasonable that if say, a course concerns only the context of real numbers, that be taken as the domain for the whole course and not have to be repeated for every individual example and exercise. | |
| Dec 29, 2015 at 23:42 | comment | added | Daniel Hast | That convention is probably a large part of why many students come out of high school with so many misconceptions and so much confusion about what functions are, which becomes readily apparent as soon as they take calculus or any higher mathematics in college. | |
| Dec 29, 2015 at 18:01 | comment | added | mweiss | @DanielHast In secondary education the standard convention is that unless otherwise specified any formula implicitly defines a real-valued function whose domain is the maximal subset of $\mathbb{R}$ for which the formula is well-defined. | |
| Dec 29, 2015 at 17:00 | comment | added | Daniel Hast | $h(x) = \frac{x}{x}$ is a formula, not a function; you haven't specified a function because you haven't given the domain and codomain of $h$. If you do specify the domain as all real $x \neq 0$, you could just as well write $h(x) = 1$ and it'd say the same thing. | |
| Dec 29, 2015 at 16:39 | comment | added | Philipp Imhof | That's a very good point indeed. | |
| Dec 29, 2015 at 16:39 | vote | accept | Philipp Imhof | ||
| Dec 29, 2015 at 16:06 | history | answered | mweiss | CC BY-SA 3.0 |