Prove that if f(x) is continuous at a then lim

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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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