Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 14 - Partial Derivatives - 14.2 Limits and Continuity - 14.2 Exercises - Page 951: 46


$f(x)= c \cdot x$ is continuous on $R^n$ when $c \in V_n$

Work Step by Step

Let us consider $c=(c_1,c_2,....,c_n)$ and $x=(x_1,x_2,...,x_n)$ ...(1) From the question, let us consider $f(x)= c \cdot x$ and $a=a_1,a_2,....a_n$ From equation (1), we have $\lim\limits_{x \to a}f(x)f(x)= \lim\limits_{x \to a}(c_1,c_2,....,c_n) \cdot \lim\limits_{x \to a}(x_1,x_2,...,x_n)$ $=c_1a_1+.....+c_na_n$ Therefore, $\lim\limits_{x \to a}f(x)=f(a)$ This shows that $f(x)= c \cdot x$ is continuous on $R^n$ when $c \in V_n$
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