Let $f:\mathbb{R} \to \mathbb{R}$ be a function and consider the difference quotient set $$D_f = \left\{\frac{f(y) - f(x)}{y-x} : (x,y) \in \mathbb{R}^2, y > x\right\}$$
Can $D_f$ be nowhere dense in $\mathbb{R}$ for non-linear $f$?
We can verify that the answer is false for $f$ continuous, since in that case $D_f$ is the continuous image of the connected set $\{y>x\}$ under $(x,y) \mapsto \frac{f(y)-f(x)}{y-x}$, hence connected (i.e., an interval) which is not nowhere dense unless $D_f$ is a singleton (which only happens for linear $f$).
But $D_f$ need not be connected in general. Indeed, consider $f = \mathbf{1}_{\mathbb{Q}}$. Then it is easy to check $D_f = \{0\} \cup (\mathbb{R} \setminus \mathbb{Q})$, which is not connected (in fact, totally disconnected).