This question is driving me nuts. I've gotten plenty of hints from office hours and emails, but I still don't quite understand it!

The questions is, of the following two statements, which implies which? Basically, which statement is stronger? The second part of the question is, does there exist a function for which both statements are true?

(a) $\displaystyle \forall \varepsilon>0, (\exists \delta>0,(\forall a\in \mathbb{R}, if |x-a|<\delta, then |f(x)-f(a)|<\varepsilon))$

(b) $\displaystyle \forall \varepsilon>0, (\forall a\in \mathbb{R}, (\exists \delta>0 if |x-a|<\delta, then |f(x)-f(a)|<\varepsilon))$

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The help that I have gotten for this problem is:

PROBLEM 2.26: You are asked to two things:

1. Tell which statement implies the other.

(Does (a) imply (b), or does (b) imply (a)?)

2. Find a function where both are true or explain why it is impossible for

both to be true.

The first thing I would is write out the statments paying attention to quantifiers, the order of quantitiers and looking for differences between the statements. Here I have translated each statement into our standard form:

a) FOR ALL epsilon > 0,

FOR ALL real numbers a,

THERE EXISTS delta > 0 such that

if |x-a| < delta, then |f(x) - f(a)| < epsilon.

b) FOR ALL epsilon > 0,

THERE EXISTS delta > 0 such that

FOR ALL real numbers a,

if |x-a| < delta, then |f(x) - f(a)| < epsilon.

Question 1 asks: If you are given a function for which (a) is TRUE,

then is it true that (b) would also be TRUE for that function?

Or in the other direction, if you are given a function for which (b) is TRUE,

then is it true that (a) would also be TRUE for that function?

Another way to say this is: which one is more specific and which one is more general? The more specific statement will also satisfy the more general statement. This all depends on the order of the quantifiers.

Question 2 asks: Can you find a function for which both are true?

The actual "if ... then ..." part of the statment is saying that

if |x-a| < delta, then |f(x) - f(a)| < epsilon

which means

if x is within a distance of delta from a,

then the output f(x) is within a distance of eplison from f(a).

In other words, this is a statement about functions where the inputs

being close together gaurantee the outputs are close together.

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When I went into office hours, he showed us that f(x)=x^2 works for the first statement but fails for the second statement whereas f(x)=x works for both statements.

I don't really know what this means. I don't understand how this problem is a "statement about functions where the inputs being close together guarantee the outputs being close together."

Thank you for any help, or even just reading the problem hhaha