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

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Lesson Parent: 
Function Limits VI: Continuous Functions
Problem: 

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

Answer: 

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

SAMPLE SOLUTION

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Solution: 

Since $f(x)$ is continuous at $a$ then by the definition of continuity we know that

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But we also know that

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and therefore

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