We say a non-constant function $f$ on $[0, 1]$ is *singular* if it is continuous, and in addition differentiable almost everywhere with $f' = 0$ a.e.

For every positive $\alpha < 1$, is the set of singular functions dense in the space of Holder continuous functions of order $\alpha$? Where we equip the space with the norm

$$\|f\|_{C^\alpha} := \sup|f| + \sup_{x, y \in [0, 1]} \frac{|f(y) - f(x)|}{|y - x|^\alpha}.$$

*Comments: The prototypical example of a singular function is the Cantor function, which is Hölder continuous of order $\frac{\log 2}{\log 3}$.*