Advantages to continuity at a point

Question

A scalar field $f : \mathbb{R}^n \to \mathbb{R}$ is said to be continuous at a point $\boldsymbol{a}$ if

$$ \lim_{\boldsymbol{x} \to \boldsymbol{a}} f(\boldsymbol{x}) = f(\boldsymbol{a}) $$

So in other words, $f$ has to be defined at $\boldsymbol{a}$ and also has to have a limit at $\boldsymbol{a}$. But isolated points are also defined to be continuous.

It seems to me like, given this definition, there isn't really an advantage to having a function continuous at a point, and continuity is only useful on an set or interval, because of the following:

• $f$ can be continuous at a point but not differentiable (e.g. $f(\boldsymbol{x}) = \|\boldsymbol{x}\|$ at $\boldsymbol{0}$)
• $f$ can be continuous at a point but the limit doesn't exist (e.g. $f(\boldsymbol{x}) = \sqrt{-\|\boldsymbol{x}\|}$ at $\boldsymbol{0}$ if $f(\boldsymbol{x}) \subset \mathbb{R}$)
• $f$ can be continuous at a point but the first-order partials don't exist

So if $f$ is continuous at a point $\boldsymbol{a}$, is there anything we can say about $f$ at $\boldsymbol{a}$? Or is it just a nice-to-have?

Answer 1

Continuity is a very important property to have, in particular when studying topology. For analysis you usually want something more, for example you'd like the function to be $C^1$ or $C^\infty$ in order to differentiate.

Also, it is nice to have a definition of continuity at a point because it shows that continuity is a local property.

Answer 2

Let's start in $R^n$ which is where you are asking the question.

I think continuity at a point is the beginning of a process. Leaving out your suggestion of isolated points, once you have defined what you mean by continuous at a point (in any number of dimensions) you can now define what you mean by continuous on an interval (or ball in $R^n$, but I'm just going to use the word 'interval'),namely that the function is continuous at every point in the interval.

Now we have something to work with. For example, a function which is continuous on a closed, bounded interval (compact set) is bounded there and is guaranteed to have a maximum and minimum there. It is also guaranteed to be uniformly continuous, which is a stronger condition. It is guaranteed that there is a polynomial on the interval which approximates the function to any desired degree of accuracy. It is (Riemann and Lebesgue) integrable on that interval and has an anti-derivative in each variable, so you have some help with integration, double integration, etc. Even if you can't figure out what this antiderivative is, the fact that it exists is quite helpful for certain proofs.

Then the entire theory of derivatives is built on top of continuity. If the function is not continuous it cannot be differentiable. In $R^n$ differentiability (as opposed to having partial derivatives, which do not even imply continuity) tell us that the curve is smoother than a merely continuous one; and that in a small enough neighborhood of any point it is nearly linear (nearly can be defined exactly). This means that if you are asked to prove something and you can prove it for linear functions -- often quite easy -- you might be able to extend your proof to differentiable functions, the argument being that nearly linear is good enough.

The existence of each additional derivative (2nd, 3rd, etc) implies that the function is smoother and smoother. This is very helpful if you are trying to approximate a function with something easier to work with or show the existence of certain things, like the solution to a differential equation.

In the complex plane the existence of a derivative even at one point is an extremely strong statement, telling you the function is analytic in some vicinity of that point. Analytic functions have many, many properties that merely differentiable ones do not, and they are the cornerstone of many solutions to problems in physics.

Getting away from $R^n$ and into other spaces, continuity can still be defined, and again continuity on a compact set has many implications -- how many and what they are depends on what space you are talking about.

So you start with continuity at a point and build a huge structure on top of that concept. As they say, the longest journey starts with but a single step.

Answer 3

Even though continuity doesn't imply differentiability, the fact that $f$ is not continuous at $\boldsymbol{a}$ means $f$ is not differentiable, and hence not continuously differentiable, at $\boldsymbol{a}$.