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The question is:

Suppose that $f$ is differentiable at a neighbourhood of $0$ and $f'$ isn't continuous at $0$. Show that there is exists sequences $x_n$ , $y_n$ such that:
$$y_n \xrightarrow{n\to\infty} 0 , x_n \xrightarrow{n\to\infty} 0.$$ Moreover: $\forall n,\ x_n \neq y_n,\ x_n \neq 0 ,\ y_n \neq 0$.
Such that:

$$\lim_{n\to\infty}\frac{f(x_n)-f(y_n)}{x_n - y_n} \neq f'(0).$$

I barely know how to begin, it's kinda trivial to show that there is exists sequence such that $0 \neq x_n \xrightarrow{n\to\infty} 0$ and $\lim_{n\to\infty} f'(x_n) \neq f'(0)$ but I don't know how does it help me, and how to continue.

I'll be glad if someone will give me a hint on how to "attack" that question.

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1 Answer 1

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Check that the problem is equivalent to finding a sequences $(x_{n})_{n}$ and $(h_{n})_{n}$ converging to $0$ such that $x_{n} \neq 0$, $h_{n} \neq 0$, $h_{n} \neq x_{n}$ and $$\frac{f(x_{n} + h_{n}) - f(x_{n})}{h_{n}} \not\to 0. \tag{$\ast$}$$

(In all my $\to$, it will be understood that I'm talking as $n \to \infty$.)


As you have noted, there exists a nonzero sequence $(x_{n})_{n}$ such that $x_{n} \to 0$ and $f'(x_{n}) \to f'(0)$. (This is because $f'$ is not continuous at $0$.)

Construct a sequence $h_{n}$ as follows: For each $n \in \Bbb N$, pick $h_{n} > 0$ satisfying the following conditions:

  1. $|h_{n}| < \frac{1}{n}$,
  2. $\left\lvert\frac{f(x_{n} + h_{n}) - f(x_{n})}{h_{n}} - f'(x_{n})\right\rvert < \frac{1}{n}$,
  3. $h_{n} \neq x_{n}$.

(The existence of such an $h_{n}$ follows essentially by the definition of the derivative.)

By Point 1., $h_{n} \to 0$. Point 3. also ensures that $h_{n}$ has the desired structure. We only need to check that $(\ast)$ holds. Note that Point 2. tells us $$\frac{f(x_{n} + h_{n}) - f(x_{n})}{h_{n}} - \frac{1}{n} < f'(x_{n}) < \frac{f(x_{n} + h_{n}) - f(x_{n})}{h_{n}} + \frac{1}{n}.$$ Now, if $\frac{f(x_{n} + h_{n}) - f(x_{n})}{h_{n}} \to f'(0)$ were true, then by Sandwich theorem, the above would give us $f'(x_{n}) \to f'(0)$, contradicting our choice of $(x_{n})_{n}$.


Note. I'm not writing $\lim_{n \to \infty} \cdots \neq f'(0)$ because that should be used when the limit does exist and does not equal $f'(0)$. However, in the above, I simply say that the limit either does not exist or does exist and does not equal $f'(0)$.

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