2

I just wanted a quick sanity check to make sure I was not misunderstanding something. In Spivak's Calculus, there is a theorem in Chapter 11 that reads as follows:

Suppose that $f$ is continuous at $a$, and that $f'(x)$ exists for all $x$ in some interval containing $a$, except perhaps $x=a$. Suppose, moreover, that $\displaystyle \lim_{x \to a}f'(x)$ exists. Then $f'(a)$ also exists, and $f'(a)=\displaystyle \lim_{x \to a}f'(x)$

Throughout this text, I have found that Spivak's word-choice is rarely redundant, which brings me to my question:

Why is it mentioned that $f'(x)$ exists for all $x$ in some interval containing $a$ if, later on, we are told that $\displaystyle \lim_{x\to a}f'(x)$ exists?

If I am told that $\displaystyle \lim_{x\to a}f'(x)$ exists, I can infer that $f'(x)$ exists for all $x$ in some interval containing $a$...though perhaps not at $x=a$..., right? (A consequence of the formal FOL defintion of the claim "$\displaystyle \lim_{x\to a}f'(x)$ exists")

I think the following is correct:

TRUE: $\displaystyle \lim_{x\to a}f'(x)$ exists $\implies$ $f'(x)$ exists for all $x$ in some interval containing $a$ (though, perhaps not for $x=a$)

FALSE: $f'(x)$ exists for all $x$ in some interval containing $a$ (though, perhaps not for $x=a$)$\implies$ $\displaystyle \lim_{x\to a}f'(x)$ exists

S.C.
  • 4,984
  • See, for instance, ŧħis. – Git Gud Nov 24 '21 at 22:35
  • 2
    The notation “$\lim_{x\to a}f'(x)$” implies that $f'(x)$ is defined in an open set/interval having $a$ as a limit point. But in my opinion it is better to recall this in plain words. – nejimban Nov 24 '21 at 22:36
  • Also it should be “$\lim_{x\to a}f'(x)$ exists and is finite” but I guess this may be implicitly understood. – nejimban Nov 24 '21 at 22:40
  • 1
    You can think of it as a stylistic choice. Instead of using symbols only, you say (or repeat) things with plain words (for example, $\sum_{i=1}^n i$ vs. "the sum of numbers from 1 to n"). Also, what $\lim_{x\to a}g(x)$ means depends on the context. In calculus, I believe the most common way is to require that $g$ be defined on some interval of $x=a$ except possibly at $x=a$. In real analysis, however, a more general definition is used where this requirement is relaxed slightly. So, you're right as long as we're talking about calculus and the standard conventions of calculus are assumed. – bjorn93 Nov 24 '21 at 23:07

1 Answers1

0

It's a bit subtle, but consider the set $S = \{\frac{1}{n} \mid n \in \mathbb{N}_+\} \cup \{0\}$. Define $f : S \to \mathbb{R}$ by $f(x) = 0$. Then the notation $\lim\limits_{x \to 0} f(x)$ makes sense when we consider the underlying space to be $S$ and not $\mathbb{R}$, and the limit is 0. This is because the $\delta-\epsilon$ definition becomes

$$\forall \epsilon > 0 \exists \delta > 0 \forall x \in S (|x - 0| < \delta \to |f(x) - 0| < \epsilon)$$

The key is that the quantifier $\forall x$ is relativised to $S$, not to $\mathbb{R}$.

So in principle, we could have some complicated function which is only differentiable one some set like $S \setminus \{0\}$ and where the limit exists.

I don't remember exactly what Spivak's conventions are, but I suspect this is the issue he was trying to avoid.

Mark Saving
  • 31,855
  • 3
    Actually Spivak does require the function to be defined in a punctured interval around the point! It's not crystal clear from his definition of limit on p. 96, but he mentions it explicitly on p. 104. – Hans Lundmark Nov 25 '21 at 07:43