I've stared at and worked with the definition of continuity of a real valued function at a point for many (like $3$) years, but there are some things that have always bothered me about it.
First, here is the definition I'm talking about:
Definition. If $f: \Bbb R \to \Bbb R$ is a function, and $x \in \Bbb R$, we say $f$ is continuous at $x$ if $\forall \epsilon > 0$, $\exists \delta > 0$ such that $|x - y| < \delta \implies |f(x) - f(y)| < \epsilon$.
I understand that this means: given any interval, or ball, around our output $f(x)$, we can always find an interval, or ball, around $x$ that is being mapped into the ball around $f(x)$.
I also understand the following two points, which I think are really important, even though they may seem obvious: 1. Given an $\epsilon > 0$, when we find some $\delta$ that satisfies this condition, any positive number smaller than $\delta$ will also work, so in effect we find infinitely many $\delta$. 2. Given an $\epsilon > 0$, if we can find a $\delta$ that satisfies this condition, then this $\delta$ works for every larger $\epsilon$, too.
So here are my questions:
We often say that intuitively, a function is continuous at a point if "small changes in the input lead to small changes in the output". But what does "small" mean? It can't mean the same thing for input and output because the changes in the input are with respect to $\delta$ and the changes in the output are with respect to $\epsilon$... who is to say these two numbers are the same type of "small"?
What is the purpose of starting the definition by looking at changes in the output (i.e., saying $\forall \epsilon > 0$ first)? If we are saying small changes in the input lead to small changes in the output, shouldn't we start with a small change in the input and check that it leads to a small change in the output?
Why do many people say, "as we shrink $\epsilon$, $\delta$ will shrink"? I don't really see how that follows from the definition. And, for example, a constant function is continuous but does not satisfy this shrinking property.