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I've been studying about limits of functions using Introduction to Analysis by Gaughan. A few days ago I asked this question Limits of Functions about the limits of functions. The motivation was curiosity about whether the idea of proving that a function has a limit at a given point could be generalized to proving that a given function had limits at all points in it's domain. After reading the next section of the book, Limits of Monotone Functions, I have another question along the same lines. In that section the author gives the following theorem and support lemma:

Lemma: Let $f : [\alpha, \beta] \to \mathbb R$ be increasing. Let $U(x) = \inf\{f(y) : x < y\}$ and $L(x) = \sup\{f(y) : y < x\}$ for $x \in (\alpha, \beta)$. Then $f$ has a limit at $x_0\in (\alpha,\beta)$ iff $U(x_0) = L(x_0)$, and in this case $\lim f(x) = f(x_0) = U(x_0) = L(x_0)$.

Theorem: Let $f : [\alpha, \beta]\to \mathbb R$ be monotone. Then $$D = \{x : x \in (\alpha, \beta)\text{ and }f\text{ does not have a limit at }x\}$$ is countable. If $f$ has a limit at $x_0 \in (\alpha,\beta)$ then $\lim f(x) = f(x_0)$.

The author goes on to say the idea is that a monotone function will have a limit everywhere except possibly at a countable set of points. My question is: if you restrict the domain of a function, as he did in the text, can this be applied to functions that aren't strictly monotone? As an example, $x^2$ is monotone on $[0, \infty)$. Based on my understanding, you could use this to show that a limit exists for all points in say $[1,3]$. While not exactly what I was looking for in my first post, this would go a long way towards that end.

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You can absolutely do as you've suggested. Once you restrict the domain of the function to a specific interval, it does not matter what happens outside that interval. As far as you are concerned, it is now a monotone function mapping $[1,3]\to \mathbb{R}$, and the theorem applies.

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To expand on AMPerrine's comment, suppose you know the theorem on countably many discontinuity points only for functions ${\mathbb R} \rightarrow {\mathbb R}.$ Then you can still apply it to the case of a monotone function $[0,3] \rightarrow {\mathbb R}$ by attaching linear extensions to the left and right of the graph of $[0,3] \rightarrow {\mathbb R}$ (which introduces at most two additional points of discontinuity) to obtain a monotone function ${\mathbb R} \rightarrow {\mathbb R}$ that you can apply the theorem to. –  Dave L. Renfro Oct 31 '11 at 13:52
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