Q. 92 - 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 $1$.  

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

Step-by-Step Solution

Answer

Ans:   $f\left(x\right)=x^{-\frac{1}{2}}$ is continuous on its domain (continuous for all $x\in R- \left\{0\right\}$.

Step 1. Given information.

given,

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

Step 2. Domain:

$f\left(x\right)=x^{-\frac{1}{2}} =\frac{1}{\sqrt{x}} =\frac{1}{\sqrt{0}} =\infty$

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

style="max-width: none; vertical-align: -60px;" $$\begin{gathered} LHS = \underset{x\to c}{lim} f\left(x\right) =\underset{x\to c}{lim} \frac{1}{\sqrt{x}} \\ Putting x=c \\ =\frac{1}{\sqrt{c}} \end{gathered}$$ 

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

Step 4. Since, LHS = RHS

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

Thus, we can write that  

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