Choose the correct answer Which of the following is a necessary condition for a function f(x) to be continuous at x=a ? f(x) must be differentiable at x=a lim_(x rarr a)f(x)=f(a) f(a)=0 f(x) must have a positive slope at x=a

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Choose the correct answer Which of the following is a necessary condition for a function  f(x) to be continuous at  x=a ?  f(x) must be differentiable at  x=a  lim_(x rarr a)f(x)=f(a)  f(a)=0  f(x) must have a positive slope at  x=a

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Recall Definition of Continuity: To determine a necessary condition for continuity at a point, we need to recall the definition of continuity at a point. A function f(x) is continuous at x=a if the following three conditions are met: 1. f(a) is defined. 2. The limit of f(x) as x approaches a exists. 3. The limit of f(x) as x approaches a is equal to f(a).Examine Given Options: Now, let's examine the given options to identify which one is a necessary condition for continuity at x=a: - ＂f(x) must be differentiable at x=a＂ is not a necessary condition for continuity because a function can be continuous at a point without being differentiable there (e.g., the absolute value function at x=0). - ＂lim_(x rarr a)f(x)=f(a)＂ directly states that the limit of f(x) as x approaches a must equal the function value at a, which is a restatement of the third condition for continuity. - ＂f(a)=0＂ is not a necessary condition for continuity; the function value at a can be any real number. - ＂f(x) must have a positive slope at x=a＂ is not a necessary condition for continuity; the function can be continuous at a point regardless of the slope or even if the slope is undefined.Identify Necessary Condition: Based on the definition of continuity and the analysis of the options, the correct answer is ＂lim_(x rarr a)f(x)=f(a)＂ because it is the only option that is a restatement of the necessary condition for a function to be continuous at a point.

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