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Do you have any tips? Especially for the second part? IS it enough to say since $f(x)$ is continuous at $x=0$ and $f$ is differentiable at $x=0$. Is $f$ continuously differentiable? Here is the question.

Let $$ f(x)= \begin{cases} x^2\sin\big(\frac{1}{x}\big) & \text{ if }x\neq 0\\ 0 & \text{ if }x= 0 \end{cases} $$ Show that $f$ is differentiable at $x=0$ and compute $f^\prime(0)$. Is $F$ continuously differentiable at $x=0$?

Edit: For the second part, I used the fundamental theorem of calculus part 2. f is continuous and according to that theorem, so F is continuously differentiable.

P.S: I don't think this question is a duplicate of another question. In that question, it is asking for the derivative of f. But, in this question, it is asking for the integral of f as the capital F is a symbol for the integral of f. I don't understand why you keep insisting that this question is a duplicate.

bebe
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  • It is certainly not enough to say that $f$ is continuous and differentiable. The question is whether the derivative of $f$ is itself continuous. (By the way, please don't mix $F$ and $f$ like this! It makes for difficult reading.) – TonyK Nov 22 '18 at 18:50
  • No in this question, they are asking whether the integral of f is differentiable or not. That is why it is capital f. And, that confused me – bebe Nov 22 '18 at 19:06
  • @TonyK I guess here they mean the integral of f. That is why is is F. This is the exact question – bebe Nov 22 '18 at 19:09
  • That question is about the derivative of f(x) but in this question, they are asking for the integral. I dont think this makes this question a duplicate. @YadatiKiran – bebe Nov 22 '18 at 19:12
  • This question is busted, then. If they are asking about the integral, they must say so. And if not, they must write $f$ instead of $F$. How can we tell where they have screwed up? (I very much doubt that they are asking about the integral of $f$. That would mean that they want to know whether the derivative of the integral of $f$ is continuous. But the derivative of the integral of $f$ is just $f$...) – TonyK Nov 22 '18 at 19:15

1 Answers1

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HINT

To check differentiability let apply the definition and check the existence of

$$\lim_{x\to 0}\frac{f(x)-f(0)}{x-0}$$

For the second point, calculate $f'(x)$ and refer to

The definition of continuously differentiable functions

user
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