Prove that if $f(x)$ is continuous at $a$ then there is a closed interval around $a$ for which $f(x)$ is bounded.

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Lesson Parent: 
Function Limits VI: Continuous Functions
Problem: 

Prove that if $f(x)$ is continuous at $a$ then there is a closed interval around $a$ for which $f(x)$ is bounded.

Answer: 

If $f(x)$ is continuous at $a$ then there is a closed interval around $a$ for which $f(x)$ is bounded.

SAMPLE SOLUTION

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Solution: 

Since $f(x)$ is continuous at $a$ we know that

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By the definition of function limits this means that for every <Sign in to see all the formulas> there exists a <Sign in to see all the formulas> such that if <Sign in to see all the formulas> then <Sign in to see all the formulas>. Furthermore, since

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we know that

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Now let's choose and arbitrary $b$ and <Sign in to see all the formulas> where <Sign in to see all the formulas>. From above we know that for every <Sign in to see all the formulas> there exists a <Sign in to see all the formulas> such that when <Sign in to see all the formulas> then

<Sign in to see all the formulas>


Since $f(a)$ is defined (by defined we mean that it is a number) and since <Sign in to see all the formulas>, when <Sign in to see all the formulas> then <Sign in to see all the formulas>. In other words, given a number $b$, we can find an interval <Sign in to see all the formulas> where <Sign in to see all the formulas>. Therefore, $f(x)$ is bounded on <Sign in to see all the formulas>.

Now we know that the open interval <Sign in to see all the formulas> contains the closed interval <Sign in to see all the formulas>. Since $f(x)$ is bounded for all $x$ in <Sign in to see all the formulas> then $f(x)$ is bounded for all $x$ in <Sign in to see all the formulas>.

This means there is a closed interval around $a$ for which $f(x)$ is bounded.