What does continuity *in general* mean?

Question

I am looking from : http://en.wikipedia.org/wiki/Lipschitz_continuity

Continuously differentiable $\subseteq$ Lipschitz continuous $\subseteq$ α-Hölder continuous $\subseteq$ uniformly continuous $\subseteq$ continuous.

What I am seeking is a precise definition of it. Intuitively, it is like I have a pointed pencil and any curve I can draw w/o lifting the pencil from the paper is continuous. BUT, what I ask is how that really boils down to the the $|f(x +\epsilon) - f(x)| < \delta $. Obviously this $\epsilon,\delta$ definition implies continuity, but is the opposite also implied?

Is there any other precise generic definition of continuity? Is the $\epsilon,\delta$ definition is the one we only have? Is that definition generic enough?

Answer 1

You have your definition all wrong. The precise definition of continuity for real functions is:

For a set $A\subseteq \mathbb R$ and a point $x_0\in A$, the function $f:A \to \mathbb R$ is continuous at $x_0$ if for every $\epsilon > 0$, there exists such a $\delta > 0$ such that for all values of $x\in A$, for which $|x-x_0|$ is smaller than $\delta$, the quantity $|f(x)-f(x_0)|$ is smaller than $\epsilon$. Written in formal language, this means: $$ \forall \epsilon>0\exists\delta>0\forall x:|x-x_0|<\delta\implies |f(x)-f(x_0)|<\epsilon $$

An extension of this definition is: if a function is continuous at every point in $A$, then it is continuous on $A$. In formal language then, $f$ is continuous iff

$$ \forall x_0\in A\forall \epsilon>0\exists\delta>0\forall x:|x-x_0|<\delta\implies |f(x)-f(x_0)|<\epsilon $$

Now, you say "this $\epsilon, \delta$ definition implies continuity", which is not really true. This definition is the definition of continuity!

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There are more definitions of continuity. For one, it is simple to prove that $f:A\to\mathbb R$ is continuous if and only if, for every convergent sequence $(a_n)$ in $A$, the equality $$f\left(\lim_{n\to\infty}a_n\right)=\lim_{n\to\infty}f(a_n)$$

Because this property is equivalent to continuity, it is an alternative definition of continuous functions.

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More generally, the concept of continuous functions can be extended to all topological spaces. There, a function is continuous if for every open set $U$, the preimage of $U$ is also open, i.e. $f:X\to Y$ is continuous if $$\forall U\subseteq Y: U\text{ open in }Y\implies f^{-1}(U)\text{ open in }X$$

Answer 2

The topological definition of continuity is what most would consider to be the most general form of continuity. It implies the $\varepsilon$-$\delta$ version of continuity in the specific example of a metric space. In a topological space, open sets give some notion of "closeness" and the topological definition of continuity says intuitively that "close points get sent to close points".

The the specific case of real valued functions, we say a function f is continuous at a point $x_0 \in \mathbb{R}$ if for every $\varepsilon > 0$ there exists a $\delta >0$ such that

$$ |x_0 - x | < \delta \implies |f(x) - f(x_0)| < \varepsilon. $$

Notice that $ |x_0 - x | < \delta$ is the same as saying $x_0 - \delta < x < x_0 + \delta$. So $f$ being continuous at $x_0$ means that if we want the output of our function $f(x)$ at some point $x$ to be close to the output $f(x_0)$ at $x_0$ (in the sense that we want $|f(x) - f(x_0)|<\varepsilon$) then we can find values of $x$ near $x_0$ ( that is $|x_0 - x | < \delta$) which guarantees $|f(x) - f(x_0)|<\varepsilon.$

To say $f$ is continuous on $\mathbb{R}$ means that $f$ is continuous at every point in $\mathbb{R}$. Intuitively again "close points get sent to close points".

Answer 3

There are some nice answers already, but I would like to highlight some other aspect. You say that continuity can be intuitively described by

I have a pointed pencil and any curve I can draw w/o lifting the pencil from the paper is continuous.

Your intuition is not wrong, but it's not a good enough (in fact you are saying that if the graph of a function is path-connected, then that function is continuous, see this question). It misses the main point of continuity (assuming $\mathbb{R}^n$ space), which is

$$\text{if the input changes a little, the output cannot change too much.} \tag{$\spadesuit$}$$

There is a big problem with the above statement, i.e. what do "little" and "too much" mean? If we would fix some relation between these changes at one point $x_0$, what about other places? The different concepts of continuity reflect different takes on the matter, and the standard $\varepsilon$-$\delta$ definition is just one of them.

Another way of saying $(\spadesuit)$ would be big change in output implies big change in input, or to make it more precise a change bigger than $\varepsilon$ in output implies a change in input bigger than $\delta$. Transposing the implication, a change bounded by $\delta$ in input implies change bounded by $\epsilon$ in output. Looking familiar?

By relating $\varepsilon$ and $\delta$ in different ways we can get different versions of continuity. If $f$ is continuous on $\mathbb{R}$, it means that, at each point there is some relation between input change and output change (input change bounds and output change bounds) and the most basic case is that (for each point) for each $\varepsilon$ there is some $\delta$.

To give you more examples, if that $\varepsilon$-$\delta$ relation is the same at all the points, we get uniform continuity. If we strengthen it further, e.g. if that $\varepsilon$-$\delta$ relation is linear, we get Lipschitz continuity (or α-Hölder continuity if the relation is like function $x \mapsto x^\alpha$). On the other hand, continuously differentiable functions are these, which change in a way so that we are able to tell, in a consistent manner, what that change is (note that $x\mapsto x^2$ on $\mathbb{R}$ continuously differentiable, but not uniformly continuous or Lipschitz).

Finally, if you get out of $\mathbb{R}^n$ into more complex spaces, there are yet different notions and definitions of continuity (the one I like the most is via nets, you can find some more info about it here), but that's not the subject of this post.

I hope that helps $\ddot\smile$

Answer 4

The $\epsilon$-$\delta$ definition of continuity is limited to metric spaces, where we have some notion of distance (the equivalent of the absolute value of a difference in $\mathbb R$). A more general definition of continuity that is for a general topological space is this:

A function $f: X \to Y$ between topological spaces is said to be continuous if for all open sets $U \subseteq Y$, the preimage $f^{-1}(U)$ is an open subset of $X$.

There are other characterizations of continuity that are equivalent with this one. When $X$ and $Y$ are metric spaces, the $\epsilon$-$\delta$ definition is equivalent to this one.

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"Obviously this $\epsilon$, $\delta$ definition implies continuity, but is the opposite also implied?"

This depends on what you take to be your definition of continuity. Remember, a definition is an if and only if statement, we just suppress the only if portion. So if the $\epsilon$-$\delta$ condition is how you define continuity, then of course continuity implies the condition, but be aware of where your definition can be applied. This is why the open set characterization of continuity is useful: because it is defined for any space where we care about continuity and it is equivalent to the $\epsilon$-$\delta$ characterization on those spaces where both can apply.