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Summary Questiontopic 2.4connecting differentiability and continuityAnswer & ExplanationDiffere

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topic 21Questiontopic 2.4connecting differentiability and continuityAnswer & ExplanationDifferentiability and continuity are two fundamental concepts in calculus that are closely related, yet distinct. Understanding their connection is crucial for analyzing the behavior of functions.Key Concepts:1.Continuity:oA function ( f(x) ) is continuous at a point ( x = a ) if:1.( f(a) ) is defined.2.The limit of ( f(x) ) as ( x to a ), denoted as ( lim_{x to a} f(x) ), exists.3.The limit equals the function value: ( lim_{x to a} f(x) = f(a) ).2.Differentiability:oA function ( f(x) ) is differentiable at a point ( x = a ) if the derivative, denoted as ( f'(a)), exists.oThis means that:The limit defining the derivative exists: $$ f'(a)=lim_{hto0}frac{f(a+h)-f(a)}{h} $$Connection Between Differentiability and Continuity:Implication: If a function is differentiable at a point, it must be c

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topic 2.4connecting differentiability and continuity




Answer & Explanation


Differentiability and continuity are two fundamental concepts in calculus that

are closely related, yet distinct. Understanding their connection is crucial for

analyzing the behavior of functions.


Key Concepts:


1. Continuity:


o A function ( f(x) ) is continuous at a point ( x = a ) if:


1. ( f(a) ) is defined.


2. The limit of ( f(x) ) as ( x \to a ), denoted as ( \lim_{x \to a}

f(x) ), exists.


3. The limit equals the function value: ( \lim_{x \to a} f(x) =

f(a) ).

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