Q. 83 - Calculus

Question

Write a delta–epsilon proof that shows that the function $f\left(x\right)=\left|x\right|$ is continuous. You may find the following inequality useful: For any real numbers $a$ and $b$, $\left|\left|a\right|-\left|b\right|\right|\le \left|a-b\right|$. (This exercise depends on Section 1.3.) 

Step-by-Step Solution

Answer

Hence we proved $\underset{x\to c}{\lim }\left|x\right|=c$.

Step 1. Given Information

We are given the function $f\left(x\right)=\left|x\right|$ is continuous and we need to write a delta–epsilon proof. 

Step 2. Delta–epsilon proof

$$\begin{gathered} f\left(x\right)=\left|x\right| \\ c=c \\ L=c \end{gathered}$$

For $\epsilon >0, \delta =\epsilon$,

If $0<\left|x-c\right|<\delta$ we have,

$$\begin{gathered} \left|\left|x\right|-\left|c\right|\right| \\ =\left|x-c\right| \\ <\delta \\ =\epsilon \\ \therefore \underset{x\to c}{\lim }\left|x\right|=c \end{gathered}$$