Cantor Function

Definition

$F(x)=\underset{n\to \infty }{\lim }{F}_n(x)$

A function $F$ that has positive derivative on Cantor Set $\mathcal{C}$ and zero derivative on $[0,1]-\mathcal{C}$

Facts

Consider the Cantor function $F$

• $F$ is a Constant Function on $[0,1]-\mathcal{C}$
• $F$ has zero derivative Almost Everywhere
• $F$ is a non-negative function

Consider the Cantor function $F$

• $F$ is monotonically increasing
• $F(0)=0$
• $F(1)=1$
• $F$ is continuous everywhere

Thus by the property of distribution function, there exists a continuous Random Variable corresponding to $F$.

$F$ has a derivative at Almost Everywhere, but the integral of the derivative and $F$ is not the same.