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Today I was thinking, given a function $f(x)$ is there any to find the domain of the function without having to do any analysis of its contents. I first thought of what a domain really represents and decided that a domain can be thought of as 'the set of all numbers that appear in the $x$ values of points on a function'*.

From there, assuming $f$ is $\Bbb R \to \Bbb R$, then if $A$ equals the set of all solutions $x$ that satisfy $$\lim_{h\to 0^+} y\{h\ge y\ge -h\}\le f(x)$$ and $B$ is equal to the set of all solutions $x$ that satisfy $$\lim_{h\to 0^+} y\{h\ge y\ge -h\}\ge f(x)$$ then would the domain of $f$ not be $A \cup B$?

*This maybe incorrect or misleading, as others have mentioned, the word 'domain' is sometimes used in incorrect ways leading to ambiguity.

** Another question related to this one.

Aaron Quitta
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  • Could you clarify what $\lim_{h\to 0^+} y{h\ge y\ge -h}$ means? – Chris Culter Sep 08 '17 at 04:10
  • It represents an a set of numbers, $y$, that are less than $h$ and greater then $-h$ as $h \to 0$ from the positive side to prevent $y$ being an empty set. – Aaron Quitta Sep 08 '17 at 04:14
  • Strictly speaking, when you write f:\mathbb R\to\mathbb R, you're defining the domain to be \mathbb R. Also, I'm not entirely what you right with the limits makes any sense, but it could just be that I personally haven't seen the notation used in that way – themathandlanguagetutor Sep 08 '17 at 04:17
  • @AaronQuitta So, $\lim_{h\to 0^+} y{h\ge y\ge -h}={0}$? – Chris Culter Sep 08 '17 at 04:18
  • Your right, I guess I meant $f(x)$ where $\forall x \in \Bbb R$ and $\forall f(x) \in \Bbb R$. The notation is probably incorrect if it doesn't look right to you, I came up with it by my self when playing with a graphical calculator not to mention my lack of formal training in this area. – Aaron Quitta Sep 08 '17 at 04:20
  • It doesn't make and sense at zero, thats why I tried to stick to the limit. – Aaron Quitta Sep 08 '17 at 04:21
  • Actually you're right, I'm being stupid. – Aaron Quitta Sep 08 '17 at 04:28
  • We all have our moments – themathandlanguagetutor Sep 08 '17 at 04:37
  • Thank you, this helped! I guess my answer is that the domain is the union of the sets A and B, where A is all x where f^-1(0) <= x and B is the set of all x where f^-1(0)>=x? This is of course assuming f has an inverse? – Aaron Quitta Sep 08 '17 at 04:43

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