Prove that $f(x)$ is continuous at $a$ if and only if \[\lim_{h\to 0}\,f(a+h)=f(a).\]

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Lesson Parent: 
Function Limits VI: Continuous Functions
Problem: 

Prove that $f(x)$ is continuous at $a$ if and only if \[\lim_{h\to 0}\,f(a+h)=f(a).\]

Answer: 

It is true that $f(x)$ is continuous at $a$ if and only if \[\lim_{h\to 0}\,f(a+h)=f(a).\]

SAMPLE SOLUTION

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Solution: 

Recall that when we say "A if and only if B" it means that we need to prove both <Sign in to see all the formulas> and <Sign in to see all the formulas>. For this particular problem we have already show that

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and that

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Therefore,

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