Why I’ll Say 1/x Is A Continuous Function And Why I’ll Say It Isn’t


So let me finally follow up last month’s question. That was whether the function “\frac{1}{x} ” is continuous. My earlier post lays out what a mathematician means by a “continuous function”. The short version is, we have a good definition for a function being continuous at a point in the domain. If it’s continuous at every point in the domain, it’s a continuous function.

The definition of continuous-at-a-point has some technical stuff that I’m going to skip this essay. The important part is that the stuff ordinary people would call “continuous” mathematicians agree with. Like, if you draw a curve representing the function without having to lift your pen off the paper? That function’s continuous. At least the stretch you drew was.

So is the function “\frac{1}{x} ” continuous? What if I said absolutely it is, because ‘x’ is a number that happens to be … oh, let’s say it’s 3. And \frac{1}{3} is a constant function; of course that’s continuous. Your sensible response is to ask if I want a punch in the nose. No, I do not.

One of the great breakthroughs of algebra was that we could use letters to represent any number we want, whether or not we know what number it is. So why can’t I get away with this? And the answer is that we live in a society, please. There are rules. At least, there’s conventions. They’re good things. They save us time setting up problems. They help us see things the current problem has with other problems. They help us communicate to people who haven’t been with us through all our past work. As always, these rules are made for our convenience, and we can waive them for good reason. But then you have to say what those reasons are.

What someone expects, if you write ‘x’ without explanation it’s a variable and usually an independent one. Its value might be any of a set of things, and often, we don’t explicitly know what it is. Letters at the start of the alphabet usually stand for coefficients, some fixed number with a value we don’t want to bother specifying. In making this division — ‘a’, ‘b’, ‘c’ for coefficients, ‘x’, ‘y’, ‘z’ for variables — we are following Réné Descartes, who explained his choice of convention quite well. And there are other letters with connotations. We tend to use ‘t’ as a variable if it seems like we’re looking at something which depends on time. If something seems to depend on a radius, ‘r’ goes into service. We use letters like ‘f’ and ‘g’ and ‘h’ for functions. For indexes, ‘i’ and ‘j’ and ‘k’ get called up. For total counts of things, or for powers, ‘n’ and ‘m’, often capitalized, appear. The result is that any mathematician, looking at the expression

\sum_{j = i}^{n} a_i f(x_j)

would have a fair idea what kinds of things she was looking at.

So when someone writes “the function \frac{1}{x} ” they mean “the function which matches ‘x’, in the domain, with \frac{1}{x} , in the range”. We write this as “f(x) = \frac{1}{x} ”. Or, if we become mathematics majors, and we’re in the right courses, we write “f:x \rightarrow \frac{1}{x} ”. It’s a format that seems like it’s overcomplicating things. But it’s good at emphasizing the idea that a function can be a map, matching a set in the domain to a set in the range.

This is a tiny point. Why discuss it at any length?

It’s because the question “is \frac{1}{x} a continuous function” isn’t well-formed. There’s important parts not specified. We can make it well-formed by specifying these parts. This is adding assumptions about what we mean. What assumptions we make affect what the answer is.

A function needs three components. One component is a set that’s the domain. One component is a set that’s the range. And one component is a rule that pairs up things in the domain with things in the range. But there are some domains and some ranges that we use all the time. We use them so often we end up not mentioning them. We have a common shorthand for functions which is to just list the rule.

So what are the domain and range?

Barring special circumstances, we usually take the domain that offers the most charitable reading of the rule. What’s the biggest set on which the rule makes sense? The domain is that. The range we find once we have the domain and rule. It’s the set that the rule maps the domain onto.

So, for example, if we have the function “f(x) = x2”? That makes sense if ‘x’ is any real number. if there’s no reason to think otherwise, we suppose the domain is the set of all real numbers. We’d write that as the set R. Whatever ‘x’ is, though, ‘x2‘ is either zero or a positive number. So the range is the real numbers greater than or equal to zero. Or the nonnegative real numbers, if you prefer.

And even that reasonably clear guideline hides conventions. Like, who says this should be the real numbers? Can’t you take the square of a complex-valued number? And yes, you absolutely can. Some people even encourage it. So why not use the set C instead?

Convention, again. If we don’t expect to need complex-valued numbers, we don’t tend to use them. I suspect it’s a desire not to invite trouble. The use of ‘x’ as the independent variable is another bit of convention. An ‘x’ can be anything, yes. But if it’s a number, it’s more likely a real-valued number. Same with ‘y’. If we want a complex-valued independent variable we usually label that ‘z’. If we need a second, ‘w’ comes in. Writing “x2” alone suggests real-valued numbers.

And this might head off another question. How do we know that ‘x’ is the only variable? How do we know we don’t need an ordered pair, ‘(x, y)’? This would be from the set called R2, pairs of real-valued numbers. It uses only the first coordinate of the pair, but that’s allowed. How do we know that’s not going on? And we don’t know that from the “x2” part. The “f(x) = ” part gives us that hint. If we thought the problem needed two independent variables, it would usually list them somewhere. Writing “f(x, y) = x2” begs for the domain R2, even if we don’t know what good the ‘y’ does yet. In mapping notation, if we wrote “f:(x, y) \rightarrow x^2 ” we’d be calling for R2. If ‘x’ and ‘z’ both appear, that’s usually a hint that the problem needs coordinates ‘x’, ‘y’, and ‘z’, so that we’d want R3 at least.

So that’s the maybe frustrating heuristic here. The inferred domain is the smallest biggest set that the rule makes sense on. The real numbers, but not ordered pairs of real numbers, and not complex-valued numbers. Something like that.

What does this mean for the function “f(x) = \frac{1}{x} ”? Well, the variable is ‘x’, so we should think real numbers rather than complex-valued ones. There no ‘y’ or ‘z’ or anything, so we don’t need ordered sets. The domain is something in the real numbers, then. And the formula “\frac{1}{x} ” means something for any real number ‘x’ … well, with the one exception. We try not to divide by zero. It raises questions we’d rather not have brought up.

So from this we infer a domain of “all the real numbers except 0”. And this in turn implies a range of “all the real numbers except 0”.

Is “f(x) = \frac{1}{x} ” continuous on every point in the domain? That is, whenever ‘x’ is any real number besides zero? And, well, it is. A proper proof would be even more heaps of paragraphs, so I’ll skip it. Informally, you know if you drew a curve representing this function there’s only one point where you would ever lift your pen. And that point is 0 … which is not in this domain. So the function is continuous at every point in the domain. So the function’s continuous. Done.

And, I admit, not quite comfortably done. I feel like there’s some slight-of-hand anyway. You draw “\frac{1}{x} ” and you absolutely do lift your pen, after all.

So, I fibbed a little above. When I said the range was “the set that the rule maps the domain onto”. I mean, that’s what it properly is. But finding that is often too much work. You have to find where the function would be its smallest, which is often hard, or at least tedious. You have to find where it’s largest, which is just as tedious. You have to find if there’s anything between the smallest and largest values that it skips. You have to find all these gaps. That’s boring. And what’s the harm done if we declare the range is bigger than that set? If, for example, we say the range of’ x2‘ is all the real numbers, even though we know it’s really only the non-negative numbers?

None at all. Not unless we’re taking an exam about finding the smallest range that lets a function make sense. So in practice we’ll throw in all the negative numbers into that range, even if nothing matches them. I admit this makes me feel wasteful, but that’s my weird issue. It’s not like we use the numbers up. We’ll just overshoot on the range and that’s fine.

You see the trap this has set up. If it doesn’t cost us anything to throw in unneeded stuff in the range, and it makes the problem easier to write about, can we do that with the domain?

Well. Uhm. No. Not if we’re doing this right. The range can have unneeded stuff in it. The domain can’t. It seems unfair, but if we don’t set hold to that rule, we make trouble for ourselves. By ourselves I mean mathematicians who study the theory of functions. That’s kind of like ourselves, right? So there’s no declaring that “\frac{1}{x} ” is a function on “all” the real numbers and trusting nobody to ask what happens when ‘x’ is zero.

But we don’t need for a function’s rule to a be a single thing. Or a simple thing. It can have different rules for different parts of the domain. It’s fine to declare, for example, that f(x) is equal to “\frac{1}{x} ” for every real number where that makes sense, and that it’s equal to 0 everywhere else. Or that it’s 1 everywhere else. That it’s negative a billion and a third everywhere else. Whatever number you like. As long as it’s something in the range.

So I’ll declare that my idea of this function is an ‘f(x)’ that’s equal to “\frac{1}{x} ” if ‘x’ is not zero, and that’s equal to 2 if ‘x’ is zero. I admit if I weren’t writing for an audience I’d make ‘f(x)’ equal to 0 there. That feels nicely symmetric. But everybody picks 0 when they’re filling in this function. I didn’t get where I am by making the same choices as everybody else, I tell myself, while being far less successful than everybody else.

And now my ‘f(x)’ is definitely not continuous. The domain’s all the real numbers, yes. But at the point where ‘x’ is 0? There’s no drawing that without raising your pen from the paper. I trust you’re convinced. Your analysis professor will claim she’s not convinced, if you write that on your exam. But if you and she were just talking about functions, she’d agree. Since there’s one point in the domain where the function’s not continuous, the function is not continuous.

So there we have it. “\frac{1}{x} ”, taken in one reasonable way, is a continuous function. “\frac{1}{x} ”, taken in another reasonable way, is not a continuous function. What you think reasonable is what sets your answer.

What I’ve Been Reading, Mid-March 2018


So here’s some of the stuff I’ve noticed while being on the Internet and sometimes noticing interesting mathematical stuff.

Here from the end of January is a bit of oddball news. A story problem for 11-year-olds in one district of China set up a problem that couldn’t be solved. Not exactly, anyway. The question — “if a ship had 26 sheep and 10 goats onboard, how old is the ship’s captain?” — squares nicely with that Gil comic strip I discussed the other day. After seeing 26 (something) and 10 (something else) it’s easy to think of what answers might be wanted: 36 (total animals) or 16 (how many more sheep there are than goats) or maybe 104 (how many hooves there are, if they all have the standard four hooves). That the question doesn’t ask anything that the given numbers matter for barely registers unless you read the question again. I like the principle of reminding people not to calculate until you know what you want to do and why that. And it’s possible to give partial answers: the BBC News report linked above includes a mention from one commenter that allowed a reasonable lower bound to be set on the ship’s captain’s age.

In something for my mathematics majors, here’s A Regiment of Monstrous Functions as assembled by Rob J Low. This is about functions with a domain and a range that are both real numbers. There’s many kinds of these functions. They match nicely to the kinds of curves you can draw on a sheet of paper. So take a sheet of paper and draw a curve. You’ve probably drawn a continuous curve, one that can be drawn without lifting your pencil off the paper. Good chance you drew a differentiable one, one without corners. But most functions aren’t continuous. And aren’t differentiable. Of those few exceptions that are, many of them are continuous or differentiable only in weird cases. Low reviews some of the many kinds of functions out there. Functions discontinuous at a point. Functions continuous only on one point, and why that’s not a crazy thing to say. Functions continuous on irrational numbers but discontinuous on rational numbers. This is where mathematics majors taking real analysis feel overwhelmed. And then there’s stranger stuff out there.

Here’s a neat one. It’s about finding recognizable, particular, interesting pictures in long enough prime numbers. The secret to it is described in the linked paper. The key is that the eye is very forgiving of slightly imperfect images. This fact should reassure people learning to draw, but will not. And there’s a lot of prime numbers out there. If an exactly-correct image doesn’t happen to be a prime number that’s all right. There’s a number close enough to it that will be. That latter point is something that anyone interested in number theory “knows”, in that we know some stuff about the biggest possible gaps between prime numbers. But that fact isn’t the same as seeing it.

And finally there’s something for mathematics majors. Differential equations are big and important. They appear whenever you want to describe something that changes based on its current state. And this is so much stuff. Finding solutions to differential equations is a whole major field of mathematics. The linked PDF is a slideshow of notes about one way to crack these problems: find symmetries. The only trouble is it’s a PDF of a Powerpoint presentation, one of those where each of the items gets added on in sequence. So each slide appears like eight times, each time with one extra line on it. It’s still good, interesting stuff.

The Intermediacy That Was Overused


However I may sulk, Chiaroscuro did show off a use of the Intermediate Value Theorem that I wanted to talk about because normally the Intermediate Value Theorem occupies a little spot around Chapter 2, Section 6 of the Intro Calculus textbook and it gets a little attention just before the class moves on to this theorem about there being some point where the slope of the derivative equals the slope of a secant line which is very testable and leaves the entire class confused.

The theorem is pretty easy to state, and looks obviously true, which is a danger sign. One bit of mathematics folklore is that the only things one should never try to prove are the false and the obvious. But it’s not hard to prove, at least based on my dim memories of the last time I went through the proof. One incarnation of the theorem, one making it look quite obvious, starts off with a function that takes as its input a real number — since we need a label for it we’ll use the traditional variable name x — and returns as output a real number, possibly a different number. And we have to also suppose that the function is continuous, which means just about what you’d expect from the meaning of “continuous” in ordinary human language. It’s a bit tricky to describe exactly, in mathematical terms, and is where students get hopelessly lost either early in Chapter 2 or early in Chapter 3 of the Intro Calculus textbook. We’ll worry about that later if at all. For us it’s enough to imagine it means you can draw a curve representing the function without having to lift your pen from the paper.

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