Q. 91 - Calculus

Question

Write a delta–epsilon proof that proves that $f$ is continuous on its domain. In each case, you will need to assume that δ is less than or equal to $1$.

  

$f\left(x\right)=x^{-2}$

Step-by-Step Solution

Answer

Ans:  $f\left(x\right)=x^{-2}$ is continuous on its domain (continuous for all $x\in R- \left\{0\right\}$)

Step 1. Given information.

given,   $f\left(x\right)=x^{-2}$

Step 2. Domain:

$f\left(x\right)=x^{-2} =\frac{1}{x^2}$

At  $x=0,$

      $f\left(0\right) =\frac{1}{0}=\infty$

Hence,  is not defined at $x=0$

Step 3. So, check for continuity at all points except 0 .

Let $c$ be any real number except $0$

assume that $c$ is less than or equal to $1$

$f$  is continuous at $x=c$

if, $\underset{x\to c}{lim} f\left(x\right)=f\left(c\right)$

$$\begin{gathered} LHS = \underset{x\to c}{lim} f\left(x\right) =\underset{x\to c}{lim} \frac{1}{x^2} \\ Putting x=c \\ =\frac{1}{c^2} \end{gathered}$$

$RHS =f\left(c\right) =\frac{1}{c^2}$

Step 4. Since, LHS=RHS

The function is continuous at $x=c (Except 0)$

Thus, we can write that 

        $f$ is continuous for all $x\in R- \left\{0\right\}$