# Understanding the notation $f\colon D\subseteq\mathbb{R}\to\mathbb{R}$ in the context of uniform continuity

In brief, I've had virtually no mathematical education prior to seven months ago. What I did know had largely been the result of watching Khan Academy. Seven months ago I began a precalculus textbook, and now I am almost done with Larson's Calculus. I love math far more than I ever would have imagined. However, every time I try to get into another textbook, something more advanced than Larson's books, I get totally flummoxed by the notation.

It seems to me that every book I can find is written for people who already know the notation; those books that do define the notation do so with great paucity of detail. In other words, if you don't know it, you're screwed. The most perplexing notation is thus

Definition 2.1 Let $f\colon D\subseteq\mathbb{R}\to\mathbb{R}$ be a function. Then $f$ is uniformly continuous if for every $\epsilon>0$, there exists a $\delta$ depending only on $\epsilon$ such that if $|x-y|<\delta$ then $|f(x)-f(y)|<\epsilon$.

I've scoured dozens of books, but none want to explain what is meant by this. The arrow symbol does not appear in lists of symbols, nor can I find a clear explanation online. This is just about the biggest hurdle preventing me from moving on to real - and hopefully very soon, complex - analysis textbooks. Please, keep in mind that I've never really had any mathematical experience beyond the books I've read. Please explain how I can interpret this symbol.

• arrow here just means $f$ maps $D$, which is defined as a subset of the reals, to a real. Congrats on educating yourself and moving above religious whackery :) – simonzack Nov 22 '15 at 15:03
• The arrow means $D$ which is a subset of $\Bbb R$ is the domain and $\Bbb R$ is the co-domain of $f$, and $f$ maps $D$ into $\Bbb R$. – user249332 Nov 22 '15 at 15:11
• en.wikipedia.org/wiki/Function_%28mathematics%29#Notation may help with the function arrow while en.wikipedia.org/wiki/List_of_mathematical_symbols may help more generally – Henry Nov 22 '15 at 15:12
• other synonyms for the symbol are: map, mapping, transformation, function, application, arrow (itself). – janmarqz Nov 22 '15 at 15:28

## 2 Answers

If we write $y = f(x)$, then $f:A\to B$ indicates that $x \in A$ and $y\in B$. It is required that $f(x)$ be defined for every $x$ in $A$ and that every value of $f(x)$ should be in $B$. Some times the relationship between $x$ and $f(x)$ is written as $x \mapsto y=f(x)$, or more commonly, as $x \mapsto f(x)$.

$A$ is called the domain and $B$ is called the codomain. $\{y\in B: \exists x\in A, y = f(x)\}$, or, less formally, $\{f(x): x\in A\}$ is called the image of $A$. It is OK for the image of $A$ to be a proper subset of the codomain of $f$. That is, $B$ needs to contain every possible value of $f(x)$, but it is allowed to be even bigger.

For example, the sine function can be thought of as a function $\sin:\mathbb R \to \mathbb R$ or as a function $\sin:\mathbb R \to [-1,1]$. Both functions have the same image but they have different codomains; so, technically, they are different functions. When the difference is important, you must explicitely state which one you mean.

The notation $f:D \subseteq \mathbb R \to \mathbb R$ is a contraction of $f:D \to \mathbb R$ where $D \subseteq \mathbb R$.

• I'd stay away from calling $B$ the range for $f\colon A\to B$; a much better term is "codomain" in my opinion. The codomain is what set you are mapping to whereas $f(A)=\operatorname{Rng}(f)$. – Daniel W. Farlow Nov 22 '15 at 16:08
• @DanielW.Farlow : I agree with you. But many high school text books do not. I find that to be very irritating. – steven gregory Nov 22 '15 at 16:10
• I don't think OP got the definition of uniform continuity from a high school text. It's pretty clear, I think, that this definition is from a book "more advanced than the pitiful Larson book." If OP is going to venture into analysis texts, then he should probably learn the correct terminology. Nonetheless, I find it doubtful that OP really "scoured" other books trying to figure out what the notation means. As you know, it's very commonplace notation. Perhaps OP wandered into a section on continuity before even looking at mappings in general. Who knows. – Daniel W. Farlow Nov 22 '15 at 16:14
• @DanielW.Farlow : You're right. And I shouldn't perpetuate the obfuscation. I will change the text. – steven gregory Nov 22 '15 at 16:17
• It's a minor point I suppose, but I kind of wonder about whether or not OP actually tried to find out what that notation meant. His "effort" seems a bit disingenuous. But kudos to you for writing up an answer. Cheers. – Daniel W. Farlow Nov 22 '15 at 16:24

You can hardly find it defined "alone" because it is part of the "complex" :

function $f$ from $A$ to $B$ (denoted : $f : A \to B$).

See :