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We have just started learning calculus, specifically about limits and continuity, and encountered a misunderstanding. Let’s consider the function $f(x) = \frac{\sin(x)}{x}$. The domain of definition is initially $\mathbb{R}^\ast$ (the real numbers excluding zero). However, by extending the function using continuity, we find that $f(0)$ exists and equals $1$.

The debate we had about the question: some of us believe the domain should now be $\Bbb R$, as we can extend the function to include zero, making it continuous at all points. Others argue that the domain remains $\mathbb{R}^\ast$ since the extension by continuity creates a new function, $\tilde{f}(x)$ which is distinct from the original.

And how does the extended function $\tilde{f}(x)$, where $\tilde{f}(0) = 1$, relate to the continuity of $f(x)$? Does this make $\tilde{f}(x)$ a new function or merely a continuation of the original one?

FShrike
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Adam
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  • Welcome to the site. While \(, \) may work for LaTex in other places, this website uses MathJax and inline equations are indicated by $ signs. For example, I had to edit: \(f(0)=1\) as $f(0)=1$ to make it render correctly – FShrike Sep 22 '24 at 12:21
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    In my opinion, the question is reasonable but very ill-posed. Who's to say the domain of definition of $f(x)=\sin x/x$ isn't $(1,2)$? Really, "$f(x)=\sin x/x$" isn't actually defining a function at all. And to actually define a function, one specifies a domain set and a codomain set, and then the question of: "what's the domain of definition?" is automatically answered. The domain of definition is whatever you want it to be, really. You could sensibly choose any nonempty subset of $\Bbb R$ for this! – FShrike Sep 22 '24 at 12:23
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    I'm being an annoying nark in my above comment, but I think that formalism is good to understand. Often in early calculus courses when people ask for the domain of definition, what they really mean is something like: "the largest 'good' set on which the function can 'make sense'" but what 'good' or 'make sense' really means is left open to interpretation. I find that annoying. TlDr in your "debate", both sides can be chosen to be correct, in my opinion; this is less mathematical and more a question of conventions – FShrike Sep 22 '24 at 12:24
  • Essentially a duplicate of https://math.stackexchange.com/q/3966934/42969. See also: https://math.stackexchange.com/q/4261834/42969, https://math.stackexchange.com/q/1743963/42969, https://math.stackexchange.com/q/462199/42969 – Martin R Sep 22 '24 at 12:42
  • @Ceru even tho you used extensions by continuity in 0 , it's in the original domain of f(x) ? – Adam Sep 22 '24 at 13:33
  • @FShrike i completely agree with u it's a less mathematical question. but let's take another example of rational functions that has in the denominator x , And you could extend the domain of that function to let 0 in it , is it mathematically correct to include it in the domain of f(x) – Adam Sep 22 '24 at 13:37

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