# Is my definition of discontinuity correct? [duplicate]

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?

• "There is a discontinuity of $\epsilon_1$..." I have never heard anyone use that phrase. Also, "For all $\delta$ around $x_0$" feels wrong too. Jun 11, 2019 at 17:27
• Alternatively, just look at the definition of continuity. If it is not met, then you have a discontinuity. They are mutually exclusive. Jun 11, 2019 at 17:28
• Possible duplicate of The negation of a limit condition, Spivak style.. The answers here are exceptional, please take a look. Jun 11, 2019 at 17:39

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 $$|f(x_0)-f(c)|\geq \epsilon_0$$.

• I have edited in accordance with what Jose said in his answer. Please take a look at the edit and say in what way it is not close
– Joe
Jun 11, 2019 at 17:41
• What do you mean by $x_0\in x?$ Or discontinuity of $a$? Pretty much, if you formulate it correctly, it will look exactly like what I wrote.
– cmk
Jun 11, 2019 at 17:43
• Shall I replace it with $x_0 \in \mathbf{R}$?
– Joe
Jun 11, 2019 at 17:44
• That would be a good start!
– cmk
Jun 11, 2019 at 17:45
• Please look at my second edit... Now is everything all right?
– Joe
Jun 11, 2019 at 17:52

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.

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.

• Please look at my edit... Now is everything all right?
– Joe
Jun 11, 2019 at 17:52
• The gist is correct. The maths is so garbled it is unusable. Jun 11, 2019 at 17:55
• Now what makes it that garbled? I have explained everything simply
– Joe
Jun 11, 2019 at 17:57