Definition of continuity without limit

Question

Without limits, I define $f(x)$ is continuous at $x=a$, when it:

1. $f(a)$ exists;
2. For every $d>0$, in the close interval $[a-d,a+d]$, there exist a maximum $M$ and a minimum $m$;
3. For every $d>0$, in the close interval $[a-d,a+d]$, for every $M>y>m$ ,there exist a $a-d≤c≤a+d$, such that $f(c)=y$.

My question is: Is this definition equal to the one with limit? How to improve it?

Answer 1

The definition you propose is not equivalent to the usual definition. Consider the function $f:\mathbb R \to \mathbb R$ given by $f(x)=x$ if $x$ is rational and $f(x)=-x$ when $x$ is irrational. With the usual definition of continuity, $f(x)$ is continuous at $x=0$. However, under your definition condition 2 fails.

You conditions also do not imply continuity. It is possible to construct non-continuous functions that satisfy the intermediate property (which is basically condition 3), such functions are called Darboux functions, and such functions can also be chosen so that condition 2 holds as well. For instance, the function $f(x)=\sin(\frac{1}{x})$ for $x\ne 0$ and $f(0)=0$ is an example of a non-continuous function at $x=0$, yet it does satisfy your conditions.