Possible alternative way of expressing continuity of a function?

Question

In Calculus or Real Analysis the usual form of definition of continuity of a function is $\epsilon- \delta$ def.

From a rigorous point of view, is it possible to say this way? and if so, why?:

$f(\lim_{n\rightarrow \infty} x_n)=\lim_{n\rightarrow \infty} (f(x_n))$

Thank you.

Answer 1

Almost. The second definition is the definition of sequential continuity. It is not hard to show that if $f$ is continuous (traditional definition) then it is sequentially continuous.

Using the Axiom of Choice, we can prove that if $f$ is sequentially continuous then it is continuous. However, the result cannot be proved without invoking at least a weak form of AC.

Answer 2

You are talking about the sequential criterion. It says:

A function $f:\Bbb R\to\Bbb R$ is continuous at $x=a$ if and only if for every sequence $a_n\to a$ we have $f(a_n)\to f(a)$.

It is essential the word 'every'. For example, take $f(x)=\sin(1/x)$. Let $a_n=1/(n\pi)$ and $b_n=1/(2n+0.5)\pi$. Both sequences tend to $0$ but $f(a_n)\to 0$ and $f(b_n)\to 1$.

Answer 3

The epsilon and the delta are understood to be numbers. For the epsilon-delta definition to be rigorous, there must be a meaning of the differences f(x)-f(a) and x-a as numbers. For example if there is a metric on the underlying spaces.

In more general cases the definition is: The inverse image of an open set is open,

Since open sets are necessary, the spaces must be topological. Otherwise there is no notion of continuity.