Suppose that $F:[a,b] \rightarrow \mathbb{R}$ is not Lipschitz in $[a,b]$, then there exists $x_0 \in [a,b]$ such that for any $\delta(x_0)>0$(delta neighbourhood of $x_0$) there is a sequence $(x_n)$ in $(x_0-\delta(x_0),x_0+\delta(x_0))$ such that $x_n \rightarrow x_0$ and the sequence $\frac{|F(x_0)-F(x_n)|}{|x_0-x_n|}$ does not converge to any real number $L$. Can somebody help me proving this? I need this to prove something.
I have examples:
$\sqrt{x}$ for $x \in [0,\infty]$ is not Lipschitz on $[0,\infty]$ and for any $\delta > 0$ neighbourhood of $0$, we can find a sequence $(x_n)$ that converges to $0$, and the sequence $\frac{|F(x_0)-F(x_n)|}{|x_0-x_n|}$ does not converge to any real number $L$.
but, I have trouble in the function $F(x)={xsin(\frac{1}{x})}\, \text{for} \, 0<x\leq 1;F(x)=0\, for \, x=0$. I know that $F(x)$ is not Lipschitz in $[0,1]$ maybe because of $0$. But, I can't find a sequence that converges to $0$ and $\frac{|F(x_0)-F(x_n)|}{|x_0-x_n|}$ does not converge to any real number $L$.