Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 2 - Review - Concept Check: 7

Answer

(a) $f$ is continuous at $a$ by definition means that $$\lim\limits_{x\to a}f(x)=f(a)$$ (b) $f$ is continuous on the interval $(-\infty,\infty)$ means $f$ is continuous at any point $x\in R$. The graph of such a function stretches from $-\infty$ to $\infty$ and is a continuous line from $-\infty$ to $\infty$.

Work Step by Step

(a) $f$ is continuous at $a$ by definition means that $$\lim\limits_{x\to a}f(x)=f(a)$$ This also means that $f(a)$ is defined and $\lim\limits_{x\to a}f(x)$ exists. In the graph, $f$ is continuous at $a$ is shown as a unbroken line at point $a$ We can intuitively think that since the line is continuous at point $x=a$, as $x$ approaches $a$, the graph of $f$ would approach continuously towards $f(a)$. So $\lim\limits_{x\to a}f(x)=f(a)$ (b) $f$ is continuous on the interval $(-\infty,\infty)$ means $f$ is continuous at any point $x\in R$. The graph of such a function stretches from $-\infty$ to $\infty$ and is a continuous line from $-\infty$ to $\infty$. According to definition, $f$ is continuous on an interval if it is continuous at any number in the interval. That means if $f$ is continuous on the interval $(-\infty,\infty)$, it is continuous at any number on $(-\infty,\infty)$, which is the whole set of rational numbers $R$.
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