I really need to get help for this problem. I'm learning continuous inverse theorem, uniformly continuous, Bolzano intermediate values theorem, and so on.

Thank you!

Originally Posted by therexists
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I really need to get help for this problem. I'm learning continuous inverse theorem, uniformly continuous, Bolzano intermediate values theorem, and so on.
It is difficult to help because we don't know how the definitions or theorems work.
How do your notes define jump? I have seen it defined exactly as what you are asked to prove.

Have you proved that if $\displaystyle f$ is increasing on open interval $\displaystyle I$ then $\displaystyle f(x-)~\&~f(x+)$ (the left-hand limit and the right-hand limit) exist for $\displaystyle \forall x\in I~?$

If you have that theorem then apply it to this problem.

If not, try to prove it. Or post the theorems and definitions you have to use.

I had to post with def and thm! But I really thank you for caring my post.
I will repost it with def and thm. I hope you will check this again.
Thanks again!