Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 1 - Limits and Their Properties - 1.4 Exercises: 59

Answer

The function has a discontinuity at each integer value of $x.$

Work Step by Step

To prove let $n$ be any integer and let $z=n-8$: $\lim\limits_{x\to n^-}f(x)=[[n^--8]]=z^-\to (z-1)\lt z^-\leq z\to$ $\lim\limits_{x\to n^-}f(x)=z-1.$ $\lim\limits_{x\to n^+}f(x)=[[n^+-8]]=z^+\to z\lt z^+\leq (z+1)\to$ $\lim\limits_{x\to n^+}f(x)=z.$ Since $\lim\limits_{x\to n^-}f(x)\ne\lim\limits_{x\to n^+}f(x)\to\lim\limits_{x\to n}f(x)$ does not exist and hence the function is not continuous at $n.$ These are non-removable discontinuities.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.