Prove that $f(x)=mx+b$ is uniformly continuous on the entire $x$-axis.

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Lesson Parent: 
Uniform Continuity
Problem: 

Prove that $f(x)=mx+b$ is uniformly continuous on the entire $x$-axis.

Answer: 

It is true that $f(x)=mx+b$ is uniformly continuous on the entire $x$-axis.

SAMPLE SOLUTION

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Solution: 

Note that <Sign in to see all the formulas> is the equation for a straight line.

We need to 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 real number $x$ and $z$ if <Sign in to see all the formulas>, then <Sign in to see all the formulas>.

Let's choose <Sign in to see all the formulas>. Let's assume that we are at an $x$ and $z$ such that <Sign in to see all the formulas>. Now let's do the following steps:

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The proof is now complete.