Q:

Can such a monstrous $f$ exist?

Call two sequences $(x_i)_{i \in \mathbb N}$ and $(y_j)_{j \in \mathbb N}$ disjoint if corresponding sets $X=\{x_1,x_2,...\}$ and $Y=\{y_1,y_2,...\}$ are disjoint.
Let $D_c$ be some set of all disjoint sequences that converge to some real number $c$ that is in the domain of $f$.
Is it possible that we have a continuous $f: \mathbb R \to \mathbb R$ and some $D_c$ such that:
a) for every sequence $(z_i)_{i \in \mathbb N} \in D_c$ we have that $\displaystyle \lim_{i \to + \infty}\dfrac {f(z_i)-f(c)}{z_i-c}$ exists (we allow it also to be $+\infty$ or $-\infty$)
and
b) if $(x_i)_{i \in \mathbb N} \in D_c$ and $(y_j)_{j \in \mathbb N} \in D_c$ are disjoint sequences then $\displaystyle \lim_{i \to + \infty}\dfrac {f(x_i)-f(c)}{x_i-c} \neq \displaystyle \lim_{i \to + \infty}\dfrac {f(y_i)-f(c)}{y_i-c}$?

A:

You could take
$$ f(x) = \begin{cases} 0 & \text{if }x=0 \\ x \sin(1/x) & \text{otherwise} \end{cases} $$
and consider the sequences
$$ z_n = \frac{1}{2\pi n+t} $$
for each $t\in[0,\pi/2]$.

