# The meaning of epsilon-delta

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$
TrueTrueTrue
TrueFalseFalse
FalseTrueFalse
FalseFalseFalse

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!

For a full in depth look on limits including an animated look, check out the following video on by 3Blue1Brown: