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I have a question regarding the definition of continuous functions : from wikipedia and my book : $f$ is said to be continuous at the point $c$ if the following holds:

For any number $\varepsilon\gt0$, however small, there exists some number $\delta\gt0$ such that for all $x$ in the domain of $f$ with $c − \delta\lt x \lt c + \delta$, the value of $f(x)$ satisfies $$ f(c) - \varepsilon < f(x) < f(c) + \varepsilon.$$

Which can be summarize by $A$ if $B$, which is the same as $B \Rightarrow A$. But then my book and wikipedia are assuming that if $f$ is continuous then we have $f(c) - \varepsilon < f(x) < f(c) + \varepsilon $:

Fix some $\varepsilon\gt0$. Since f is continuous, there is a $δ > 0$ such that $|f(x) − f(c)| < \varepsilon$ whenever $|x − c| < δ$. This means that $$ f(x) − \varepsilon < f(c) < f(x) + \varepsilon.$$

Which is the same than assuming $A\Rightarrow B$ but our definition of continuity only states that $B\Rightarrow A$. Is there something that I don't get ? Or is my memory of mathematical logic wrong ?

Thanks in advance.

Cameron Buie
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user1594047
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1 Answers1

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The definition of a predicate can be expressed as a bi-conditional :

$f$ is continuous at point $c$ iff for all $\epsilon > 0$, there exists some $\delta > 0$ such that for all $x$ in the domain of $f$, if $c − \delta\lt x \lt c + \delta$, then $f(c) - \varepsilon < f(x) < f(c) + \varepsilon$.

Thus, in order to prove the continuity of $f$ at $c$, we have to choose a positive $\epsilon$ whatever (for all $\epsilon > 0$) and we have to find a suitable positive $\delta$ (there exists $\delta > 0$) such that, for any $x$, the condition "if ..., then ..." is satisfied.

In this way, we "use" the right-to-left part of the bi-conditional :

if for all $\epsilon > 0$, ..., then $f$ is continuous at point $c$.

I think that part of the doubt is due to the "correct formalization" of the defining clause.

You can see here a discussion about the conditional statements and the formalization of "if and only if".

The usual way to state the above definition is to omit the iff and states it as :

$f$ is continuous at point $c$ if ...

This means that we are saying : "$A$, if $B$", that is we are using the part : "if $B$, then $A$" of the bi-conditional, i.e. :

if for all $\epsilon > 0$ ..., then $f$ is continuous at point $c$.

Why so ? because "usually" we have to apply the defining condition in order to prove the continuity.

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Mauro ALLEGRANZA
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  • Then why it is not written in my book and on wikipedia f continous iff [...] but it is written f continous if [...] ? – user1594047 Nov 16 '14 at 18:16
  • @user1594047 For some reason (I guess it is tradition) to write only "if" and not "iff". But definitions are if and only if statements; even if doesn't say so. – UserB1234 Nov 18 '14 at 03:43