how to prove $f^\prime(x)$ is not bounded? $f$ is a differentiable function in $(0, 1)$. The limit $\lim_{x \to 0^+}f(x)$ doesn't exist. Prove that $f^\prime(x)$ is not bounded in $(0, 1)$.
I can see why it's true, but I am struggling proving it.
I hope you guys can help me. Thank you!

Put $\alpha=\limsup_{x \to 0^{+}} f(x), \beta = \liminf_{x \to 0^{+}} f(x)$.
We may choose strictly decreasing sequences $a_n, b_n \to 0^{+}$ with $a_n \neq b_n$ for each $n$, and with $f(a_n) \to \alpha, f(b_n) \to \beta$. Note the former condition implies $a_n - b_n \to 0$. Hence $$\limsup_{n \to \infty} \left|\frac{f(a_n) - f(b_n)}{a_n - b_n}\right| = \limsup_{n \to \infty} \overbrace{|\alpha - \beta|}^{>0}\left|\frac{1}{a_n - b_n}\right| = +\infty$$
and the result follows from the MVT.
