We have this function
$$f(x) =
\begin{cases}
x^2, & \text{if $x$ $\in \mathbb{Q}$} \\[2ex]
0, & \text{if $x$ $\in\mathbb{I}$}
\end{cases}$$
We have to determine where the function is differentiable and calculate the derivative. Then we need to determine where is not differentiable and prove this with $\epsilon – \delta$
Attempt:
We define:
$$\text{$h(x)=0$ and $g(x)=x²$, then $h(0)=0=g(0)=f(0)$}$$
Now we have:
$$\text{$h(x)$ ≤ $f(x)$ ≤ $g(x)$, for all x$\in\mathbb{R}$}$$
$$\text{that is equal to: $h(x)-h(0)$ ≤$ f(x) – f(0) $≤$ g(x) – g(0)$}$$
$$\text{And without losing generality assuming that $x>0$:}$$
$$\text{$\frac{h(x)-h(0)}{x}$ ≤ $\frac{f(x)-f(0)}{x}$ ≤ $\frac{g(x)-g(0)}{x}$}$$
$$\text{Finally: $\lim_{x\to 0} \frac{h(x)-h(0)}{x}$ ≤ $\lim_{x\to 0} \frac{f(x)-f(0)}{x}$ ≤ $\lim_{x\to 0} \frac{g(x)-g(0)}{x}$}$$
$$\text{By the squeeze theorem we have that:}$$
$$\lim_{x\to 0} \frac{f(x)-f(0)}{x}=0$$
$$\text{So $f$ is differentiable at $c=0$ and $f'(0)=0$}$$
The problem comes when we have to show that is not differentiable at any other point. I managed to do it with limits but I don't know how to put in into $\epsilon – \delta$
$$\text{Let $x\in\mathbb{Q}$ and $x≠0$. Then $f(x)=x²$. Now we know that exist a sequence $(y_n)_{n\in\mathbb{N}}$ of irrational numbers such that:}$$
$$\lim_{n\to \infty}y_n = x$$
$$\text{Also we have that $f(y_n)=0$ for all $n$, because $y_n$ is irrational, but:}$$
$$\lim_{n\to \infty}f(y_n) = f(x) = x²$$
$$\text{So we can see that:}$$
$$\lim_{n\to \infty}f(y_n) ≠ f(y_n)$$
$$\text{This implies that $f$ is not continuous at $\mathbb{I}$, therefore $f$ is not differentiable at $\mathbb{I}$}$$
$$\text{The same thing works to prove that $f$ is not continuous at $\mathbb{Q}$ so $f$ is not differentiable at any point, except $x=0$.}$$
As I said the problem comes when I have to write that second thing with $\text{$\epsilon – \delta$ }$, cause I don't really know how to start. Someone has any ideas? Thanks to everyone.
Best Answer
Allow $(x_n)$ be a sequence of rational numbers such that $x_n \to \sqrt{2}$.
Fix $\epsilon > 0$. First notice that $|x_n - \sqrt{2}| < \delta \implies |x_n + \sqrt{2}| = |x_n - \sqrt{2} + 2\sqrt{2}| \leq |x_n - \sqrt{2}| + 2\sqrt{2} < \delta + 2\sqrt{2}$.
So $|x_n - \sqrt{2}||x_n + \sqrt{2}| < \delta(\delta + 2\sqrt{2})$.
Allowing $\delta < \frac{\epsilon}{1 + 2\sqrt{2}}$ for small enough $\epsilon$ (i.e. if $\epsilon < 1 + 2\sqrt{2}$) produces
$|f(x_n) - 2| = |x_n-\sqrt{2}||x_n + \sqrt{2}| < \delta(\delta + 2\sqrt{2}) < \delta(1+2\sqrt{2}) < \frac{\epsilon}{1+2\sqrt{2}}(1+2\sqrt{2}) = \epsilon$
Thus, $\lim_{x_n \to \sqrt{2}} f(x_n) = 2$, however $f(\sqrt{2}) = 0$.
It is not continuous, thus not differentiable.