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Chapter 3: Problem 8

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

Short Answer

Expert verified
Answer: Yes, if a function is differentiable at a point, it implies that the function is continuous at that point.

Step by step solution

01

Definition of Differentiability

A function \(f\) is said to be differentiable at a point \(a\) if the derivative \(f'(a)\) exists. This means that the limit $$\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$$ exists.
02

Definition of Continuity

A function \(f\) is said to be continuous at a point \(a\) if the following three conditions are met: 1. \(f(a)\) exists. 2. \(\lim_{x\to a}f(x)\) exists. 3. \(\lim_{x\to a}f(x) = f(a)\).
03

Connection between Differentiability and Continuity

Now let's investigate the relationship between differentiability and continuity. If \(f\) is differentiable at \(a\), it means that $$\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$$ exists. To demonstrate that \(f\) is continuous at \(a\), we will show that the limit $$\lim_{x\to a} f(x) = f(a)$$.
04

Change of Variable

We can use a change of variable to connect the derivative limit to the limit of the function value. Let \(h = x - a\), and as \(x\) approaches \(a\), \(h\) approaches \(0\). We want to compute the limit $$\lim_{x\to a}\frac{f(x)-f(a)}{x - a} = \lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$$.
05

Apply Differentiability

Since \(f\) is differentiable at \(a\), we already know that the above limit exists and is equal to \(f'(a)\). Therefore, we have $$\lim_{x\to a}\frac{f(x)-f(a)}{x - a} = f'(a)$$.
06

Multiply by the Denominator and Apply Limit Rules

Next, we can multiply both sides of the equation by the denominator \((x-a)\) and utilize limit properties to obtain $$\lim_{x\to a}(f(x)-f(a)) = f'(a) * \lim_{x\to a}(x-a)$$. As \(x\) tends to \(a\), the right side of the equation becomes \(f'(a) * 0 = 0\). Therefore, $$\lim_{x\to a}(f(x)-f(a)) = 0$$.
07

Evaluate the Limit

As a result, we have \(\lim_{x\to a} f(x) - f(a) = 0\). Adding \(f(a)\) to both sides yields \(\lim_{x\to a} f(x) = f(a)\).
08

Conclusion

Since \(\lim_{x\to a} f(x) = f(a)\), we can conclude that if a function \(f\) is differentiable at a point \(a\), it must also be continuous at that point.

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