1

In the answer to my question here, it is asserted that when $f:[0, \infty)\to\mathbb{R}$ is a function defined as $f(x) = \sqrt{x}$, it is valid & precise to write: $$\lim_{x\to 0}f(x) = 0$$ because in the answer to my question here, $\lim_{x\to c}f(x) = L$ only requires that $c$ be a limit point. This is in agreement with Rudin's definition of $\lim_{x\to p}f(x) = q$.

Now when discussing discontinuity, Rudin asserts that $\lim_{t\to x}f(t)$ exists if and only if $f(x+) = f(x-) = \lim_{t\to x}f(t)$. But, as far as I can understand from Rudin's definition of $f(x-)$, which is the left-hand limit, when $f(x) = \sqrt{x}$, $f(0-)$ does not exist, and therefore, $\lim_{t\to 0}f(t)$ does not exist as well. But, this contradicts the previous paragraph that says that $\lim_{t\to 0} \sqrt{x}$ exists because $0$ is a limit point.

What do I miss?

Community
  • 1
Tadeus Prastowo
  • 273
  • 2
    You could say the left hand limit exists because for all sequences $x_1,x_2,\ldots$ of points such that $x_i<0$ and $x_i$ is in the domain, you have $f(x_i) \to 0$. It just happens that no such points exist. – J126 Nov 24 '15 at 12:10
  • Interesting!!!! What a very subtle trick Rudin has!!! For all sequences, and there again, a vacuous truth! – Tadeus Prastowo Nov 24 '15 at 12:21
  • 1
    I totally disagree with saying that the left-hand limit exists in the case of $\sqrt{x}$. But this is not a problem in the situation that Rudin is describing, since the very first thing he says in Definition 4.25 is "Let $f$ be defined on $(a,b)$", and then "It is clear that at any point $x$ of $(a,b)$, $\lim_{t\to x} f(t)$ exists if and only if [...]". So he's not talking about continuity at the endpoints here, only about continuity at interior points. – Hans Lundmark Nov 25 '15 at 14:02
  • @HansLundmark: A very good observation. Yes, Rudin is defining left- and right-hand limits in a specific context, which is more specific than the more general context in which Rudin defines the more general definition of limit in Definition 4.1. – Tadeus Prastowo Nov 25 '15 at 15:42

0 Answers0