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Theorem 3.10 in Tom Apostol's Calculus states that for each strictly increasing and continuous function $f$ in the interval $[a, b]$, it's inverse function $g=f^{-1}$ is continuous on $[c, d]$, where $c=f(a), d=f(b)$.

Theorem 3.8 in the same book states that every continuous function takes all values in it's range. In the above example, that means that $f$ takes all values in $[c, d]$ interval and that $g$ takes all values in $[a, b]$ interval.

To demonstrate my understanding on a simple example. Let $f(x)=10x$ for $x \in [0.1, 1]$. Then the continuous inverse of $f$ is $g(y)=\frac{y}{10}$ for $y \in [1, 10]$. From the theorems above we know that each number from the interval $[0.1, 1]$ maps to a number in $[1, 10]$ interval, and vice versa.

I'm having troubles understanding the implication of those two theorems in reality, because it seems that it makes all real intervals equal in length, which I find so hard to fit into the reality, that I doubt real numbers might not be real. For example, if the real numbers are real, do the above theorems imply that if one stick is 11m long, and another one 1.1m long, then the middle 9m of the first stick and the middle 0.9m of the second stick should be equal in length?

Thanks!

S11n
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1 Answers1

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If by size you mean cardinality, then yes. But cardinality only measures a particular kind of size. In the same sense that $\mathbb{N}$ has the same cardinality as $\mathbb{Z}$.

In general it is a bit pointless to consider "are the real numbers even real". It comes down to pointless philosophical arguments with no actual substance. Also, the middle of your stick is a point, which in general has no size.

EDIT: I see you clarified that with length you mean size. An invertable function just tells you two sets have the same size in the sense of cardinality. In the sense of measure, the existence of a continuous invertable function between two subsets of $\mathbb{R}$ sais nothing about the underlying measures.

Stefanos Van Dijk
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  • Cardinality concept is quite new to me, but I think the difference between your example and this one is that N and Z are both infinite, and it seems that a stick long 9m has a finite length. I didn't say the middle point, but I just wanted to avoid having some ambiguity about one sided continuity. It probably doesn't matter, but I did anticipate someone might think it's a pointless question, so I wanted to avoid pointless answers too, so I added that...Currently I do not find the question pointless, but maybe I will as I continue learning. – S11n Jul 05 '21 at 14:51
  • @S11n What concept of size are you using? Because the two intervals most definatelly do not have the same measure. – Stefanos Van Dijk Jul 05 '21 at 16:00
  • Thanks for the comment Stefanos! Unfortunately I might be trying to combine uncombinable things - length of a real interval and length of a stick (as a helping tool building the intuition). The most rational description of the stick length I can think of is the number of atoms in it, which would actually be exactly the cardinality, and most probably the two sticks would have different count of atoms. But, that itself points out that I cannot think of a real description in which I would use a real length of the stick (e.g. real intervals have real length). – S11n Jul 06 '21 at 05:47
  • Maybe a simpler way to approach this is to ask if a irrational numbers are observable (as a subset of real numbers)? For example, I saw people thought for a long time if $\pi$ was irrational or not. Then I saw someone thought that because $\pi$ was irrational that could be the reason why circles could not exist in reality etc...so I guess I wonder if any of those questions make sense or I should just take mathematics as an idealized abstraction and not try to much to bother with the real word analogies at this level (I'm just a beginner, as you can see)? – S11n Jul 06 '21 at 05:51