BlizzardWalker

Anyway, if you think to keep it, then let it be in accordance with your regulations. I have nothing to say about that! Thanks in advance for your fast comments on the requests, appreciate all the hard work you guys do!

I did not say I cheated, I did not know it was a publicly lively contest. My apologies in advance but all I asked was to delete it, I didn't intend to receive such harsh criticism

@XanderHenderson Thank you for the fast response. The reason is this question was posted from a lively math contest which still ongoing so deleting this question would be a very valid reason imo.

There has been pretty a lot of downvotes for this problem, if any of users could delete this question, it would be super appreciated!

There have been pretty a lot of downvotes for this problem and this problem is from the ongoing live contest, so if any of the moderators could delete this question, it would be super appreciated!

@MartinR Do you think the solution for the original problem is correct?

Who will be the next president of the United Mathematics Stack Exchange

How is going on everybody?

it is just more mathematical rigor that is lacking

t

I am pretty sure that your logic is correc

thank you so much!

I agree with the well-definition of $x_{\epsilon}$. But, how do you get that $x_{\epsilon}$ approaches to $a$ whenever $\epsilon$ approaches to zero?!

Honestly, the last part really lacks the mathematical rigor: how $f(a) = 0$ and continuity of $f$ directly implies that limit of $x_{\epsilon}$ approaches to $a$ whenever $\epsilon$ approaches to zero: This really requires mathematical formalism, intuitively this seems okay though (however, I am still confused on how you choose these $\epsilon$ values, as this is not correct to assume that you have the control of the choices of these $\epsilon$ values.

Oh, now I see the point. But now arises another question: how continuity of $f$ will guarantee that for any small values of $\epsilon$, you can find that the function values near the neighbourhood will be larger than $\epsilon$? I guess there should be another point $y_0$ if we are dealing with continuity

Moreover, you obtained that $x_{\epsilon}$ is larger than one quantity. How can you take the limits for each $\epsilon$ where you do not have control over the choices of $\epsilon$

Can you please exemplify more on the set of $x_{\epsilon}$?! I still can not figure out the meaning of this set. (pretty sure this comes from continuity of $f$, but I would be really appreciated if you could provide a little more details if possible)

So it means that $x_{\epsilon}$ is the smallest ("infimum") number for which $x_{\epsilon} > a$ and $|f(x_{\epsilon})| >= \epsilon$? Meaning that for all numbers $y$ less than $x_{\epsilon}$, we will know that $|f(y)|$ is less than $\epsilon$?

I guess the equality of $f(a) = 0$ could be derived by just plugging $x=a$ in the given inequality

$$ |f(x_\varepsilon)| \le K\int_{a}^{x_\varepsilon}|f(t)|dt \le K(x_\varepsilon-a)\varepsilon $$ How do you derive the left-hand-side inequality?!

How do you know that the maximum of function $g$ will be epsilon?! Instead of epsilon, there should be the supremum of the function $g$ on that interval (i.e. absolute value of $f$)

I do not understand exactly why $1 <= K(x - a)$ implied that $x = a$

how is going on?!

well hello!

Which cruel evil has chopped me off from the chat

@porridgemathematics How is going in your neighbourhood?

Hello

hey

What

@matt NOOOOO

hahaha

who is deusovi

yeo

I am winner

I will see when moderators will move you out of this forum

"Always provide proper attribution for all copied material." Aha, so you basically wanna rule me out, I see

really be ashamed of yourself

@D.W. I am not as proficient as you, but I know ALL THE RULES IN THIS FORUM, and IF I DID NOT MENTION SOURCE, THEN THERE IS SOME REASON THAT I DID NOT DO IT!!

@D.W. Next time be more careful when you pick words

@D.W. He told me not to mention his name here!!!

@D.W. And also, downvoting gives you nothing but hatred feelings towards you! Really sad that this forum has such unfriendly users, who are very eager to downvote any amateur's posts.

@D.W. I have every right to ask a question in this forum, providing in the statement all my findings and progress. What else do you want from me?

@D.W. This is original problem created by my professor.

@D.W. And he allowed me to post this question in this forum

@D.W. What is problem with posting this question?