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Pragya, You are correct that differentiability can be defined on a closed interval by using one sided limits. Some calculus text books do introduce one sided derivatives but many do not. Most of the applications of differentiation apply to two sided derivatives and as a result authors don't take the additional effort to introduce one sided derivatives. So why go to the extra effort to define continuity from the left and from the right in order to define continuity on a closed interval? The reason is that many of the important theorems in calculus are theorems about continuous functions on a closed interval. Three of these theorems are The Extreme Value Theorem, The intermediate Value Theorem and The Mean Value theorem. Penny  


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