Prove that $f(x)=1/x$ is uniformly continuous on the interval $[1,\infty)$.

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Lesson Parent: 
Uniform Continuity
Problem: 

Prove that $f(x)=1/x$ is uniformly continuous on the interval $[1,\infty)$.

Answer: 

It is true that $f(x)=1/x$ is uniformly continuous on the interval $[1,\infty)$.

SAMPLE SOLUTION

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Solution: 

We should note that we have proven that this is continuous (not uniformly continuous) on <Sign in to see all the formulas>.

To prove this 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> 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> and see if this works. Now let's assume that we are at an $x$ and $z$ in the interval <Sign in to see all the formulas> such that <Sign in to see all the formulas>.

Because we are restricting ourselves to the domain <Sign in to see all the formulas> we know that <Sign in to see all the formulas> and <Sign in to see all the formulas>. By multiplying <Sign in to see all the formulas> by $z$ we have <Sign in to see all the formulas> and by multiplying <Sign in to see all the formulas> by $x$ we have <Sign in to see all the formulas>. Therefore, <Sign in to see all the formulas>. Now let's do the following steps:

<Sign in to see all the formulas>

But we have assumed that $x$ and $z$ are such that <Sign in to see all the formulas> and we know that

<Sign in to see all the formulas>

Therefore,

<Sign in to see all the formulas>

and the proof is now complete.