Is my definition of discontinuity correct?

Question

I think I have made a rigorous and intuitive definition of discontinuity.

Definition: For a function $f(x)$ from $\mathbf{R}$ to $\mathbf{R}$, there is a discontinuity of $a$ at $x_0 \in \mathbf{R}$ if for all $\delta > 0$ around $x_0$; whenever $|x-x_0|<\delta$, there exists an $x$ such that $|f(x)-f(x_0)|$ $\geq a$

How much rigor is my definition?

Answer 1

Your definition is kind of close, if I'm being generous. To get a full, rigorous notion of discontinuity, you just need to negate the definition of continuity: a function is discontinuous at $x_0$ if for some $\epsilon_0>0$ and any $\delta>0,$ there exists $c$ so that both $|x_0-c|<\delta$ and $|f(x_0)-f(c)|\geq \epsilon_0$.

Answer 2

There is zero rigor, because:

1. “For all $\delta$ around $x_0$” means nothing.
2. You don't say whar $x_0$ is.
3. The expression “the greatest value of $\lvert f(x)−f(x_0)\rvert$” is undefined.
4. The number $\varepsilon_1$ is undefined.

Answer 3

Your definition doesn’t quite work; “largest” may not exist (what if there is an infinite sequence approaching a bound, for example). Using maximum won’t work, you’d need to use supremum. You could simply replace “greatest value of” with “there exists an $x$ such that”, and then your definition works.