The epsilon-delta definition for limits states that (from Wikipedia) for all real $\epsilon > 0$ there exists a real $\delta > 0$ such that for all $x$ with $ 0 < |x − c | < \delta$, we have $|f(x) − L| < \epsilon$ – however, the definition of the limit requires only the existence of some $\delta>0$ for any $\epsilon>0$. The part I am having trouble understanding is why there are no details as to the "intuitive" decrease of the δ as ε grows smaller.
I realize that saying that as ε approaches zero δ also approaches zero would use the non-rigorous intuition of a limit in a definition meant to make the limit a rigorous part of mathematics, but why is it unnecessary to show the relationship between epsilon and delta besides the proof of existence? Is there some other implication of a function that I am missing that is the reason only the proof of existence is in this definition?
EDIT: If we consider the dependence of on an epsilon on a decreasing delta, can the limit exist if epsilon is increasing as delta decreases? If so, why?
Best Answer
For the purposes of limits, the precise dependence of $\delta $ on $\epsilon$ is simply not important. The proofs do not require any knowledge of that relation. As said, $\delta (\epsilon)$ usually tends to $0$ as $\epsilon$ tends to $0$, but this is not always the case. To make this precise, let $\delta(\epsilon)$ be the largest $\delta$ corresponding to $\epsilon$ in the definition of continuity at $x$ for a function $f$. Thus, $\delta(-)$ is a function whose domain is $(0,\infty )$ and whose range is $(0,\infty]$, and it is monotonically non-decreasing. If $f$ is a constant function, then $f(\epsilon)=\infty $ for all $\epsilon>0$, showing that indeed $\lim_{\epsilon\to 0}\delta(\epsilon)$ need not be $0$.
The definition of limit captures the following: $\lim_{x\to a}f(x)=L$ means that for any prescribed distance $\epsilon>0$, there exists some upper bound for distances $\delta$ such that if $x\ne a$ is within $\delta $ units from $a$, then $f(x)$ is guaranteed to be within $\epsilon$ units from the limit $L$.
Remark: The function $\delta(-)$ above is known as a modulus of continuity for $f$. Functions whose moduli of continuity have certain properties (e.g., are concave) are of importance.