# Epsilon-Delta or Infinitesimal: Which is More Rigorous?

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• observer1
In summary, the author is discussing the use of informal terminology, such as "infinitesimal," to avoid discussing the complexities of rigor in integration concepts. There are two main concepts where the ##\varepsilon-(\delta)-## construction is used: limits and continuity. These concepts can be explained as the closer you get with one thing, the closer you get with another. The index ##\varepsilon## indicates the chosen accuracy in both cases. Continuity means that no matter how close function values are at a certain point, there is always a small range of ##x##-values around this point where all the function values are close to the function value at that point. Overall, the use of infinitesimals means achieving
observer1
Background: mechanical engineer with a flawed math education (and trying to make up for it).

I have recently read this statement (and others like it): "We shall also informally use terminology such as "infinitesimal" in order to avoid having to discuss the (routine) "epsilon-delta" analytical issues that one must resolve in order to make these integration concepts fully rigorous."

Could someone elaborate on what this means?

I the cloud of my memory, I see all these words as ONE CONCEPT on a path toward learning integration.

Were there actually TWO ways I was taught, which, over the years, I got all jumbled up in my head? Is one more rigorous that the other? What does the author mean when he puts the word "routine" in parentheses (they are his, not mine).

There are mainly two concepts, where the ##\varepsilon-(\delta)-##construction is used: limits (including differentials) and continuity (of functions). Both can informally be explained as "the closer you get with one thing, the closer you get with another".

In the case of limits it is "the more you proceed (get close) to infinity, the closer you get to the limit".
Formally this means $$\lim_{n \rightarrow \infty} a_n = a \Longleftrightarrow \;\forall \;\varepsilon > 0 \;\exists\; N_\varepsilon \in \mathbb{N} \;\forall \;n > N_\varepsilon \;|a_n-a|<\varepsilon$$
Literally it is: Given any positive amount of chosen accuracy (##\varepsilon##), there is a number (##N_\varepsilon##), from which on (##n>N_\varepsilon##) all elements of the sequence (##a_n##) are within that margin (##|a_n - a| < \varepsilon##) to the limit (##a##).

In case of continuity it is "no matter how close function values at a certain point are, there is always a small range of ##x-##values around this point, where all of their function values are such close to the function value in that given point".

Formally this means

A function ##\displaystyle f\colon D\to \mathbb {R}## is continuous in a point ##\displaystyle \xi \in D ##, if for each ##{\displaystyle \varepsilon >0}## there is a ## {\displaystyle \delta_\varepsilon >0}##, such that for all ##{\displaystyle x\in D}## with ##{\displaystyle |x-\xi |<\delta_ \varepsilon}## holds ##{\displaystyle |f(x)-f(\xi )|<\varepsilon }##.

It is a bit tricky not to confuse the quantifiers. An easy way to remember is, to consider a step function, which is not continuous at the step ##\xi##. In this case you will always find points ##x## around ##\xi## that don't lead to arbitrary close function values, no matter how close you chose the area of possible ##x-##values around ##\xi##.

In both cases the index ##\varepsilon## indicates, that the choices of ##N_\varepsilon## and ##\delta_\varepsilon## depend on the chosen value of ##\varepsilon\,##, the "accuracy" if you like. Infinitessimal now only means "with an accuracy as close to a zero error margin as you like".
You can even easily draw pictures of the situation, because you don't have to bother the scaling: the situation is the same on small as on larger scales. Of course only until no other point of discontinuity is in the way. However, as long as you don't consider pathological examples, you don't need to worry, whether ##\varepsilon = 0.0001## or ##\varepsilon = 0.5## to draw a picture.

## What is the difference between epsilon-delta and infinitesimal?

Epsilon-delta is a method of evaluating limits in calculus by using a variable (epsilon) to determine how close the input value (delta) must be to the limit. Infinitesimal, on the other hand, refers to an infinitely small value that is used in non-standard analysis to represent a number that is smaller than any real number.

## Which method is more commonly used in calculus?

Epsilon-delta is the more commonly used method in calculus because it is based on the standard definition of limit and is accepted by most mathematicians.

## Why was there a historical controversy between these two methods?

There was a historical controversy between epsilon-delta and infinitesimal because they have different interpretations of the concept of "infinitely small." Epsilon-delta is based on the concept of a limit approaching a fixed value, while infinitesimal is based on the concept of a number that is smaller than any real number.

## Can epsilon-delta and infinitesimal give different results when evaluating limits?

Yes, epsilon-delta and infinitesimal can give different results when evaluating limits. This is because they have different definitions of what it means for a limit to exist. Epsilon-delta requires the limit to approach a fixed value, while infinitesimal allows for the limit to approach different values depending on the infinitesimal value used.

## Are there any real-life applications of epsilon-delta and infinitesimal?

Both epsilon-delta and infinitesimal have real-life applications in different fields of mathematics. Epsilon-delta is commonly used in calculus to evaluate limits, while infinitesimal is used in non-standard analysis to study infinite and infinitesimal numbers. Additionally, both methods have been applied in physics and engineering for the study of continuous and discrete systems.

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