What is the interpretation of this mathematical statement? Identify whether true or false:
If $f(x)=(x-1)\{x\}$, where $\{x\}$ denotes the fractional part of $x$, the limit of $f(x)$ does not exist at all integers.
This is how I went about it:
The limit of $f(x)$ exists for $x=1$. However, it doesn't exist for any other integers. So, it doesn't exist for $2$, doesn't exist for $3$, etc.
Hence, the limit doesn't exist for all integers. The statement is true.
Here's how my teacher went about it:
The limit of $f(x)$ exists for $x=1$. However, it doesn't exist for any other integers. Hence, it exists for a single integer, i.e. not all integers. Hence, the statement is false.
Which interpretation is correct and which one is wrong (as there can only be one answer)?
 A: The difference between the two possible interpretations is whether the $\forall$ quantification ought to be included in the negation or not. And indeed it is very different to say “not everywhere does this happen” than to say “everywhere, this doesn’t happen”! Without any additional information, both interpretations of the wording of the statement are valid.
However, you should know that it is very common practice (though not a very good one) in almost all areas of mathematics to push quantifiers after the predicate: e.g.
$$\cos2x = \sin^2x - \cos^2 x, \qquad \forall x \in \mathbb R. $$
This is because in spoken language the correct order of the logical elements is reversed. But everyone knows that this should be understood as
$$\forall x \in \mathbb R\quad \cos2x = \sin^2x - \cos^2 x.$$
So the instructions, whenever you see a quantifier where it shouldn’t be (after the predicate), would be to automatically pull it back where it belongs. In your specific situation, this means that “the limit does not exist | at all integers” should be reworded as “at all integers | the limit does not exist.” This is your teacher’s interpretation.
Another point in favor of your teacher’s version is that your interpretation should be formalized as:
$$\neg (\forall x\in\mathbb Z \quad \text{limit exists at }x ),$$
which is equivalent to saying
$$\exists x\in\mathbb Z \quad \neg(\text{limit exists at }x),$$
or, in common parlance, “there is an integer $x$ such that the limit does not exist at $x$.” Because of this “switch” property of quantifiers (“not everywhere does this happen” means “somewhere, this doesn’t happen,” i.e. $\neg\forall = \exists\neg$, and viceversa) it is more common to push the negation through the quantifier and have its complement modify the negation. All in all, if your interpretation was the one intended by the author of the exercise, then they would likely have written “the limit does not exist for some integer,” or more precisely “for some integer, the limit does not exist.”
A: Letting $P(x)$ be "$\lim f(x)$ at $x$ exists", the teacher's interpretation is
$$\forall x\in\mathbb Z\ \neg P(x)$$
while your interpretation is
$$\neg\forall x\in\mathbb Z\ P(x)$$
However, the given emphasis on at all integers makes it more likely that the teacher's interpretation is intended. If not, the question would have been reworded.
