Prove that x2 is uniformly continuous on the domain [−10,10].

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Lesson Parent: 
Uniform Continuity
Problem: 

Prove that x2 is uniformly continuous on the domain [−10,10].

Answer: 

It is true that x2 is uniformly continuous on the domain [−10,10].

SAMPLE SOLUTION

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Solution: 

We must show that for every <Sign in to see all the formulas> there exists a <Sign in to see all the formulas> such that for any x and z in <Sign in to see all the formulas>, then if <Sign in to see all the formulas>, then <Sign in to see all the formulas>.

Let's make a guess of <Sign in to see all the formulas>. Furthermore, let's assume that x and z are in <Sign in to see all the formulas> and that <Sign in to see all the formulas>.

We can use this rule to show that

<Sign in to see all the formulas>


Since <Sign in to see all the formulas> we know that

<Sign in to see all the formulas>


Now since x and z are assumed to bin in <Sign in to see all the formulas>, the most that |x+z| can be is 20. Therefore,

<Sign in to see all the formulas>


So when <Sign in to see all the formulas>, then <Sign in to see all the formulas>. When <Sign in to see all the formulas>, then <Sign in to see all the formulas>. Therefore, no matter what <Sign in to see all the formulas> is we have chosen a <Sign in to see all the formulas> so that when <Sign in to see all the formulas>, <Sign in to see all the formulas>.

The proof is now complete.