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

Answer: 

It is true that if $f(x)$ is continuous at $a$ then there is an open 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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Since <Sign in to see all the formulas> can be as small as we want, we choose a <Sign in to see all the formulas> so that

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Since $f(a)$ is defined, when <Sign in to see all the formulas>, <Sign in to see all the formulas>. Therefore, $f(x)$ is bounded on <Sign in to see all the formulas>.