If $f$ is a differentiable function with $f'$ discontinuous at $0$, show that the set of all real $y$ 's such that $\lim_{n\rightarrow \infty}f'(x_n)=y$, where $x_n$ is a real sequence such that $\lim_{n\rightarrow \infty} x_n = 0$, is equal to the segment $[p,q]$ for some $p\not = q$ (can be infinity).
I'm wondering if this problem states the valid thing... Because if we take $f(x)=\ln x$ then, indeed, $f'$ is discontinuous at $0$ but if we take any $\{x_n\}: x_n\rightarrow 0,n\rightarrow\infty$, then $\lim_{n\rightarrow\infty}f'(x_n)$ is either $+\infty$ or $-\infty$ not a segment (even with infinity bounds as the problem allows)