Q. 89 - 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 in its domain $x\in R$.

Step 1. Given information.

given, 

           $f\left(x\right)=x^2$

Step 2. Domain

Since $x$ is defined for 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 $1$

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

 $LHS = \underset{x\to c}{lim} f\left(x\right)=\underset{x\to c}{lim} x^2$

 by putting $x=c$

         $=c^2$

 $RHS =f\left(c\right) =c^2$

Step 4. Since, LHS =RHS

So, Function is continuous at $x=c$

Thus, we can write that 

                   $f\left(x\right)=x^2$ is continuous for all $x\in R$