I started to write a short note about the limit and continuity some days ago. The question occured when I read some of the proofs, for example this <link> where the two limits were assumed to exist. Since they exist the formal definition of a limit would be applied and then using them to prove the goal. Now, the question is:

Let $f:A\to \mathbb{R}$ be a function. Do we say

$f(x)\to L$ as $x\to c$, if $$\forall \epsilon>0\exists\delta>0:\forall x\in A: 0<|x-c|<\delta \Longrightarrow |f(x)-L|<\epsilon $$

like many books say or this <link>, or do we say

$f(x)\to L$ as $x\to c$, if and only if $$\forall \epsilon>0\exists\delta>0:\forall x\in A: 0<|x-c|<\delta \Longrightarrow |f(x)-L|<\epsilon $$

like this <link>? I think the second one makes sense when it comes to the proofs.


Use "if and only if" if you want to be precise. The reason why many people use "if" in definitions is because they are not mathematically precise, partly due to the English "if" having the ability to function as definitional and not just conditional. For example:

You can go out if you finish your homework.

which implies that you cannot go out if you don't finish your homework.

However in mathematics "if" is usually taken with the conditional meaning and the above deduction would be invalid. So one should use "if and only if" or "iff" for short when we want equivalence.

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