On negation of lipschiz continuity Let $f: [a,b] \to R$ be continuous function which is not Lipschitz continuous.
Can we say there exist $x \in [a,b] $ and  strictly monotone sequences, $\{x_n\}_{n=1}^{\infty} \subseteq [a,b] $ and $\lambda_{n} \in \mathbb{R^+} $ such that $x_n \to x$ and $\lambda_{n}  \to + \infty $,  
$$|f(x_n) - f(x)| > \lambda_{n}  |x_n - x|  $$ 
for all $n \in \mathbb{N}$? 
P.S: Cleary we don't need to care about monotonicity of those two sequences! 

No, this is false. Consider a function defined on $[0,1]$ that mainly follows the line $y = 0$, but as $x \to 0$ has infinitely many increasingly steep peaks up to the line $y = x$.
(Formally, the function is zero, except on intervals $[1/n - 1/3n^2, 1/n + 1/3n^2]$, where it is piecewise linear, consisting of a triangle with vertices at the points $(1/n - 1/3n^2,0)$, $(1/n,1/n)$ and $(1/n + 1/3n^2,0)$.)
