The meaning of epsilon-delta

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

limxx0f(x):=ϵ>0,δ>0,[xx0<δf(x)f(x0)<ϵ]\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:

limxx0f(x):=\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

limxx0\lim_{x \to x_0}

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

ϵ>0\forall \epsilon > 0

Oh it’s for all ϵ>0\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 δ\delta:

δ>0\exists \delta > 0

Together:

ϵ>0,δ>0\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:

[xx0<δf(x)f(x0)<ϵ]\Bigl[|x - x_0| < \delta \Rightarrow |f(x) - f(x_0)| < \epsilon \Bigr]

The sentences

xx0|x-x_0|

and

f(x)f(x0)|f(x) - f(x_0)|

should be clear, as we are just taking the difference of xx and x0x_0, and f(x)f(x) and f(x+0)f(x+0). The absolute value tells us that these two sentences are talking about distance. Well, we know x0x_0 and f(x0)f(x_0) are just numbers, and they are constant. So this phrase must be talking about the distance of xx to x0x_0 and the distance of f(x)f(x) to f(x0)f(x_0), or simply, the distance between the input relative to x0x_0 vs. the distance between the output relative to f(x0)f(x_0)! Then:

xx0<δf(x)f(x0)<ϵ|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

xxyyx    yx \implies y
TrueTrueTrue
TrueFalseFalse
FalseTrueFalse
FalseFalseFalse

Putting it all together, we see that this line says when the distance of the input relative to x0x_0 is small, the distance of the output relative to f(x0)f(x_0) is small.

xx0<δ    f(x)f(x0)<ϵ|x - x_0| < \delta \implies |f(x) - f(x_0)| < \epsilon

How small? Arbitrarily small! If ab<ϵ|a - b| < \epsilon for all ϵ>0\epsilon > 0, then a=ba = b! The only way xx0|x-x_0| can be small is if xx was actually really close to x0x_0. That is the purpose of delta! To measure how close we are to x0x_0. So when we move close to x0x_0, we should expect to move closer to f(x0)f(x_0)!

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

limxx0f(x):=ϵ>0,δ>0,[xx0<δf(x)f(x0)<ϵ]\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 xx and x0x_0 is smaller than delta, we can make the distance between the output f(x)f(x) and f(x0)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!

##Further Reading

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

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Zhi Han