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 $1$.

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

Short answer

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

Step-by-step solution

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, role="math" localid="1648053742246" $\underset{x\to c}{lim} f\left(x\right)=f\left(c\right)$

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

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

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$x\in R$.