What would happen if we changed the quantifiers in a continuity of a function?

[loop quantum gravity]

TL;DR

continuous function.

The definition of a continuous function at a point $x_0$ is:
$$\forall \epsilon >0 \exists \delta >0 \forall x(|x-x_0|<\delta \rightarrow |f(x)-f(x_0)|<\epsilon)$$

What would happen if we change it to another defintion as: $\exists \epsilon >0 \forall \delta>0 \forall x(|x-x_0|<\delta \rightarrow |f(x)-f(x_0)|<\epsilon)$?
or instead for all $x$ change it to there exists an $x$ s.t what is in parantheses is satisified.
How would we see it graphically?

There are other permutations of the quantifiers and the "if then" connective which we can look at, and replace it with conjunction.

 

[Roberto Pavani]

Your question is not clear to me.

$\exists \epsilon >0 \forall \delta>0 \forall x(|x-x_0|<\delta \rightarrow |f(x)-f(x_0)|<\epsilon)$

It seems to me that this would classify as continuous a function like:

$f(0)=0$
$f(x) = 1$ for $x \neq 0$

 

[Matterwave]

In your new definition, $\epsilon$ is only required to be a positive number so all you're saying is the function is bounded.

The $\forall \delta>0 \forall x$ just makes the conditional inside "trigger" sometimes and not other times. When the antecedent ($|x-x_0|<\delta$) is true, it just says $f$ is bounded. When the antecedant is false, nothing is stated.

Try it with any bounded function and see that it satisfies your new definition. Try it with the Dirichlet function.

 

[loop quantum gravity]

In your new definition, $\epsilon$ is only required to be a positive number so all you're saying is the function is bounded.

The $\forall \delta>0 \forall x$ just makes the conditional inside "trigger" sometimes and not other times. When the antecedent ($|x-x_0|<\delta$) is true, it just says $f$ is bounded. When the antecedant is false, nothing is stated.

Try it with any bounded function and see that it satisfies your new definition. Try it with the Dirichlet function.

How about other variations of the quantifiers and perhaps also changing sometimes the logical connective from implication to conjunction?
Usually the definition in books for bounded functions is as such:
$$\forall M>0 \forall x (|f(x)|<M)$$
I can see indeed what you mean that it may be a bounded function or not; and with conjunction above it would be exactly boundedness.

 

[Matterwave]

$$\forall M>0 \forall x (|f(x)|<M)$$

I think you mean $\exists M$

I'm not sure what you're trying to get at with swapping quantifiers in this expression. I can't give some general rule for what happens. Try them out yourself and see. Most likely you don't get very useful statements.

 

[loop quantum gravity]

yes there exists an M such that...
Anyways I thought it to be an interesting logical-calculus exercise to understand every subtelty. I guess I can ask an AI for further assistance. Thanks though!!