Problem 73
Question
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If a function is differentiable at a point, then it is continuous at that point.
Step-by-Step Solution
Verified Answer
The statement is True.
1Step 1: Understanding the Definitions
The first step is to understand the definitions of continuity and differentiability. A function \(f(x)\) is continuous at some point \(c\) if the limit of the function at \(c\) equals to the value of the function at \(c\). A function is said to be differentiable at a point \(c\) if the derivative \(f'(c)\) exists.
2Step 2: Apply The Theorem
The theorem in calculus states that if a function is differentiable at a point, then the function must be continuous at that point. However, the converse is not necessarily true: a function may be continuous at a point but not differentiable.
3Step 3: Evaluating the Statement
According to the theorem, the given statement that 'If a function is differentiable at a point, then it is continuous at that point.' is indeed True.
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