Q. 93 - 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 $x\in R$)

Step 1. Given information.

Given,  $f\left(x\right)=x^{1/2}$

Step 2. Domain:

Since, $x$ is defined fo all $x\in R$ 

Step 3. So, check for continuity.

Let $c$ be any real number.

$f$ is continuous at $x=c$

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

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

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

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

Step 4. Since, LHS=RHS

So, Function is continuous at $x=c$

Thus, we can write that 

       $f\left(x\right) =x^{\frac{1}{2}}$ continuous for all $x\in R$.