I'm reading about differentiation in textbook Analysis I by Amann.
.
Consider $f:[0,1] \to \mathbb R$. Clearly, $0$ is a limit point of $[0,1]$.
Assume that $\lim _{x \rightarrow 0^+} \frac{f(x)-f(a)}{x-a}$ exists. Then $\lim _{x \rightarrow 0} \frac{f(x)-f(a)}{x-a} = \lim _{x \rightarrow 0^+} \frac{f(x)-f(a)}{x-a}$ exists. This is because any sequence $(x_n)$ in $[0,1]$ such that $x_n \to 0$ contains only $x_n \ge 0$. As such, $f$ is differentiable at $0$ according to Prof. Amann.
Usually, other textbooks said that $f$ does not have derivative at $0$, but only has the right derivative at $0$. So I'm very confused.
I would like to ask if my understanding is correct! Thank you so much!