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

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Lesson Parent: 
Function Limits VI: Continuous Functions
Problem: 

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

Answer: 

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

SAMPLE SOLUTION

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Solution: 

We know that

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and therefore

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The function limit

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means that $f(x)$ is continuous at $a$ by the definition of continuity.