Q. 90 - 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 $1$.  

$f\left(x\right)=x^3$

Step-by-Step Solution

Answer

Ans:   $x^3$ is continuous in its domain $x\in R$

Step 1. Given information.

given expression  $f\left(x\right)=x^3$

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 $1$

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

$LHS=\underset{x\to c}{lim} f\left(x\right)=\underset{x\to c}{lim} x^3$

by putting $x=c$

        $=c^3$

$RHS = f\left(c\right)=c^3$

Step 4. Since, LHS=RHS

So, Function is continuous at $x=c$

Thus, we can write that 

         $f\left(x\right)=x^3 continuous for all x\in R$