Epsilon-delta vs. infinitesimal

[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).

 

[fresh_42]

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.