Problem 3 - Calculus Early Transcendentals

Question

If \(f\) is differentiable at \(a,\) must \(f\) be continuous at \(a ?\)

Step-by-Step Solution

Answer

Answer: Yes, if a function is differentiable at a point, it must be continuous at that point. This is because differentiability implies the existence of a limit of the difference quotient, which in turn implies continuity at the given point.

Step 1: Defining continuity at a point

A function \(f\) is continuous at \(a\) if: 1) The limit of the function as x approaches a exists: \(\lim_{x\to a} f(x)\) exists 2) \(f(a)\) is defined 3) \(\lim_{x\to a} f(x) = f(a)\)

Step 2: Defining differentiability at a point

A function \(f\) is differentiable at \(a\) if its derivative exists at that point, which is denoted as \(f'(a)\). In other words, the limit of the difference quotient as x approaches a must exist: $$\lim_{x\to a} \frac{f(x) - f(a)}{x-a}$$

Step 3: Relationship between continuity and differentiability

We have the definitions of continuity and differentiability in front of us. The key to understanding their relationship lies in differentiability. If a function is differentiable at a point, it means that the limit of the difference quotient as x approaches that point must exist. Moreover, if this limit exists, it implies that the function must also be continuous at that point. This is because, if a function is not continuous, its limit might not exist, contradicting the differentiability condition.

Step 4: Conclusion

If a function \(f\) is differentiable at a point \(a\), it must be continuous at that point too. This is because, for differentiability, the limit of the difference quotient must exist, and this can only happen if the function is continuous at that point. Therefore, the answer to the question is yes: if a function is differentiable at a point, it must be continuous at that point.