The meaning of epsilon-delta

15 Sep, 2018

5 min read

Hi, I’m Zhi, and this is my first post at Less Than Epsilon! This will be a website for posting math and physics related content. At the time of posting, I am just beginning my third year of undergraduate studying Mathematical Physics at the University of Alberta.

Although this is a visual blog filled with pictures, however being able to read mathematics without any pictures is a really useful skill.

The motivation for the name comes from the epsilon-delta definition of continuity, which is

\lim_{x \to x_0} f(x) := \forall \epsilon > 0, \exists \delta > 0,\\ \Bigl[|x - x_0| < \delta \Rightarrow |f(x) - f(x_0)| < \epsilon \Bigr]

Let’s break down this formula. The left hand side says:

\lim_{x \to x_0} f(x) :=

where the $:=$ symbol means we are defining a new concept. In this case, we are defining the limit operator

\lim_{x \to x_0}

In this case the function is $f$ , which the input is $x$ and the output is $f(x)$ . Writing $f$ for function can be ambiguous, so usually it’s better to write $f(x)$ . Since I didn’t say what $f(x)$ had to be, so it can be any function.

\forall

The first symbol we see is the for all symbol. It’s an upside down A, A meaning for all. It means, the following expression must be true, for all what?

\forall \epsilon > 0

Oh it’s for all $\epsilon > 0$ ! Now what? Here we see another symbol:

\exists

Which means there exists. This symbol asserts the existence of something. We see that it asserts the existence of a $\delta$ :

\exists \delta > 0

Together:

\forall \epsilon > 0, \exists \delta > 0

Okay so for every epsilon we have a delta. Now what. The next line tells us what we do with these epsilons and deltas:

\Bigl[|x - x_0| < \delta \Rightarrow |f(x) - f(x_0)| < \epsilon \Bigr]

The sentences

|x-x_0|

and

|f(x) - f(x_0)|

should be clear, as we are just taking the difference of $x$ and $x_0$ , and $f(x)$ and $f(x+0)$ . The absolute value tells us that these two sentences are talking about distance. Well, we know $x_0$ and $f(x_0)$ are just numbers, and they are constant. So this phrase must be talking about the distance of $x$ to $x_0$ and the distance of $f(x)$ to $f(x_0)$ , or simply, the distance between the input relative to $x_0$ vs. the distance between the output relative to $f(x_0)$ ! Then:

|x - x_0| < \delta \\ |f(x) - f(x_0)| < \epsilon

Means these distances are smaller than epsilon and delta. A inequality like this can either be true or false. That’s where the last symbol comes in:

\implies

The implies symbol means “if”. If it is raining, then I will bring an umbrella. If I have an umbrella, did it rain? Not necessarily, I could have brought

$x$	$y$	$x \implies y$
True	True	True
True	False	False
False	True	False
False	False	False

Putting it all together, we see that this line says when the distance of the input relative to $x_0$ is small, the distance of the output relative to $f(x_0)$ is small.

|x - x_0| < \delta \implies |f(x) - f(x_0)| < \epsilon

How small? Arbitrarily small! If $|a - b| < \epsilon$ for all $\epsilon > 0$ , then $a = b$ ! The only way $|x-x_0|$ can be small is if $x$ was actually really close to $x_0$ . That is the purpose of delta! To measure how close we are to $x_0$ . So when we move close to $x_0$ , we should expect to move closer to $f(x_0)$ !

Putting it all together, we see that the definition of a limit is:

\lim_{x \to x_0} f(x) := \forall \epsilon > 0, \exists \delta > 0,\\ \Bigl[|x - x_0| < \delta \Rightarrow |f(x) - f(x_0)| < \epsilon \Bigr]

Which means for every epsilon greater than 0, we can find a delta greater than 0, such that when the distance between the input $x$ and $x_0$ is smaller than delta, we can make the distance between the output $f(x)$ and $f(x_0)$ smaller than epsilon, or as one might say, less than epsilon. Anyways that is where the name comes from. Thanks for reading!