Can difference quotient sets be nowhere dense?

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 certainly false for $f$ continuous, since continuous functions map connected sets to connected sets.