Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question

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.

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.