One might expect, perhaps naively, that mathematical definitions should be interpreted as "iff" statements by default, unless stated otherwise. This expectation arises from the facts that a definition expresses an equivalence, and that equivalences are biconditional.

But the more I read mathematics textbooks, the more it seems that the distinction between "if" and "iff" is not respected. Consider for example the following definition from Elementary Linear Algebra, 5ed, by Larson, Edwards, and Falvo (Houghton Mifflin Co.).

My question is: Why are those ifs not iffs instead? Are the authors just being sloppy?

It's supposed to be iff. I suppose it's just tradition or something. I've been mentally substituting "iff" for a long time, so I no longer notice when "if" is used.

That brings me to my next question: How do you know when you can make the substitution?

Consider this example from Calculus, 8ed, by Larson, Hostetler, and Edwards (Houghton Mifflin Co.).

In that definition, it is supposed to be "if". Otherwise you would conclude that the function [itex]f(x)=\sqrt{1-x^2}[/itex] is not continuous at [itex]x=\pm 1[/itex] because the limit doesn't exist at either of those points.

It's pretty normal to use "if" in mathematical definitions. It's more along the lines of how "if" is used in everyday speech than in logic, such as "a motorcycle is called a crotch rocket if it goes over 225 km/h and causes severe back pains". You don't then call a harley a crotch rocket.

Sadly that was the only example that's coming to me right now.

This is a sloppy definition if it's supposed to encompass the case of endpoints. I would hope they seperately mention the case of endpoints.

Of course they may have already mentioned something about the meaning of a "two sided" limits at points where the function is defined on only one "side", so this could still be included (I'm not sure I've ever seen an interpretation like this though).

They define continuity on a closed interval (with its attendant one-sided limits) a couple of pages later.

This was first brought to my attention a year ago when a student asked me about the continuity of a function like [itex]f(x)=\sqrt{1-x^2}[/itex] at [itex]x=\pm 1[/itex]. He mentally substituted "iff" in for "if" in the definition I quoted, and concluded that [itex]f(x)[/itex] is not continuous at the endpoints. I pointed out that he shouldn't be using the two-sided limit. I subsequently wrote up his argument as a "find the flaw in this reasoning" problem and have been assigning it ever since.

No problem with the book then, that definition of continuity is for points contained in an open interval that's contained in the domain of f. Your student tried to apply that definition where it didn't apply.

The calc book I don't have much of an issue with. I do think it should make a point of the use of "if" in that definition, but since they don't, I do it in class. The student was definitely the one in error, but the mistake was a subtle one.

But I do think that the linear algebra book should have used iff in the definition of a diagonalizable matrix.

There's a horrible implicit thing being applied here: if you're claiming [itex]f(x) := \sqrt{1 - x^2}[/itex] is a function, then you had better be working in the topological space [-1, 1]. If you're working in the topological space [-1, 1], [itex]\lim_{x \rightarrow 1} f(x)[/itex] does equal f(1). Since there does not exist points greater than 1, the ordinary "two-sided" limit at 1 is exactly the same thing as the left one-sided limit.

And, if you were using some generalized definition of function so that this f was a "function" on the reals, then I would agree with your student that said this function is not continuous at 1. (when working over the reals) Of course, I'm assuming what it would even mean for such a generalized function to have a limit.

I would still maintain it's a different usage of the word "if" than you find in logic say. To elaborate on what I mean with a less silly example, we call a tree deciduous "if" it sheds it's leaves before winter. Would you call a pine tree deciduous? In the same way I wouldn't call a matrix that's not similar to any diagonal matrix diagonalizable if I had read your books definition.

It's also the language that can be found in definitions by many respected authors, Titchmarsh, Rudin, Lang, Royden, and speaking of a gold standard in linear algebra, Hoffman & Kunze. In fact I had to do a bit of searching on my shelf before I found a definition that used "iff", that was Ahlfor's complex analysis which oddly used "iff" for the definition of a compact space but none of the other definitions I looked at. Maybe because it was followed by a few "iff" equivalences of compact that he had iff on the brain.

Regarding the limit, I'd prefer if they looked at limits as being in whatever topological space makes sense for the domain as Hurkyl is talking about, but this isn't the kind of viewpoint you're likely to get from a "stock" calculus text.

The problem was indeed framed that way. The exact problem statement was:

which is similar enough to the [itex]f(x)[/itex] that I used.

That part is news to me. Would I find it in a book on analysis? Topology? Either?

Fortunately it hasn't been an issue, because I steer them away from applying this definition of continuity for functions on closed intervals anyway.

You may have agreed with the conclusion, but you certainly would not have accepted his argument. He said that the function is discontinuous at the endpoints because the limit doesn't exist there. Taking the definition exactly as it appears in the textbook, the student definitely committed the fallacy of denying the antecedent.

But even if we replace "if" with "iff", the argument still fails because the limit does in fact exist.

Yes, but in the "real world" example there is common sense at work. What I am wondering is when we should do that in mathematics.

Just to put this into perspective: My graduate education is in theoretical physics, but I am currently teaching 1st and 2nd year undergrad math (intro linear algebra and below). I was not sure of how much liberty I could take with my interpretation of definitions. More specifically, I was under the impression that I could take no liberties with them. So I held that "if" means the logical "if", and nothing else. But now I have a better picture of things, I think.

OK, so it's normal. That's one of the things I was trying to find out.

I would prefer it too. I will definitely be reading up on this and finding a way to teach it in a calculus course.

There has to be some agreed upon English (or other language) or you are going to have trouble communicating the math. You could probably do it, but it would involve a lot of grunting and frantic waving

About the limit thing, you're just changing the "if 0<|x-a|<delta" part of the limt to "if 0<|x-a|<delta and x is in the domain of f". have a look at how limits are defined in metric spaces. You'd just be considering the closed interval a metric space, and you can't "see" points outside this interval when looking at limits.

In single variable calc, they do everything in terms of left and right limits. This is a pretty natural thing to do on the real line, but having a more general notion of the limit introduced in single variable would probably make the jump to higher dimensions seem a little smaller.

I would argue that the limit does, in fact, not exist.

The problem here is that we've not defined our terms! We all learned in calculus the definition of the limit of a function at a point. But we have not (at least I have not) ever seen anyone define the limit of a partial function[/b] at a point.

I presume that our difference in opinion is that we have different opinions on the "proper" definition of the limit of a partial function.

The definition I had in mind of [itex]\lim_{x \rightarrow a} f(x) = L[/itex] (on the reals) is:

[tex]
\forall \epsilon > 0 : \exists \delta > 0 : \forall x \in (a - \delta, a) \cup (a, a + \delta) : |f(x) - L| < \epsilon
[/tex]

Thus, the expression [itex]\lim_{x \rightarrow 1} \sqrt{1 - x^2}[/itex] is ill-defined because we can never choose [itex]\delta[/itex] so that [itex]f(x)[/itex] is known to exist.

I'm not saying this is the right way to do this; better ideas spring to mind, but I'm not entirely sure I like their consequences either.

No, I'm talking about the problem that I quoted. We were restricted to the space between the endpoints. You said that in that case, the limit as x approaches an endpoint exists and is equal to the value of the function there.

It would really depend on how the book defined [tex]\lim_{x\rightarrow a} f(x)[/tex]. It's typical for a calculus text to use the usual epsilon-delta way that makes no limitations of the "x" being in the domain of f(x) (what Hurkyl gave in post 18). However, when they do this they will (or should!) explicitly state that this definition of the limit is for values of a where f is defined on an open interval containing a.

This is a little bad in the sense that for some deltas you may be considering "invalid x" values, but if delta is small enough you're ok given that f is defined in a neighbourhood of a, so it's not a huge deal.

But your text gave a different definition for continuity at an endpoint so they likely will only talk about one sided limits at endpoints.