Q. 88 - 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. 

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

Step-by-Step Solution

Answer

Ans:  $f\left(x\right)=x^{-1}$ is continuous on its domain (continuous for all $x\in R- \left\{0\right\}$)

Step 1. Given information.

given, $f\left(x\right)=x^{-1}$

Step 2. Find Domain.

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

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

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

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

Step 4. L H S = R H S

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

Thus, we can write that

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