Prove that if $f(x)$ is continuous at $a$ and $f(a)\neq 0$ then $1/f(x)$ is continuous at $a$.

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Lesson Parent: 
Function Limits VI: Continuous Functions
Problem: 

Prove that if $f(x)$ is continuous at $a$ and $f(a)\neq 0$ then $1/f(x)$ is continuous at $a$.

Answer: 

It is true that if $f(x)$ is continuous at $a$ and $f(a)\neq 0$ then $1/f(x)$ is continuous at $a$.

SAMPLE SOLUTION

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Solution: 

By the definition of continuity

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Furthermore, by the assumption of the problem we know that

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Therefore, we know that

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And again by the definition of continuity, <Sign in to see all the formulas> is continuous at $a$.