Prove that $f(x)=x^2$ is not uniformly continuous on the entire $x$-axis.

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Lesson Parent: 
Uniform Continuity
Problem: 

Prove that $f(x)=x^2$ is not uniformly continuous on the entire $x$-axis.

Answer: 

It is true that $f(x)=x^2$ is not uniformly continuous on the entire $x$-axis.

SAMPLE SOLUTION

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Solution: 

To prove that $x^2$ is not uniformly continuous we must first state the negation of uniform continuity. A function $f(x)$ is not uniformly continuous on a set $U$ if there exists an <Sign in to see all the formulas> such that for every <Sign in to see all the formulas> there are two numbers $x$ and $z$ in $U$ where <Sign in to see all the formulas> and <Sign in to see all the formulas>.

Let's choose <Sign in to see all the formulas>, <Sign in to see all the formulas> and <Sign in to see all the formulas>. Now let's assume we have any arbitrary <Sign in to see all the formulas>. First, we have

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Second, we have

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So we have shown that <Sign in to see all the formulas> and <Sign in to see all the formulas>. The proof is now complete.