A function that is not a distribution function
Explain why F(x)=0 for x\le 0 and F(x)=1 for x>0 is not a distribution function. Change one inequality so that it becomes one.
Worked answer (click to show)
The proposed F has F(0)=0 and F(0+)=1, so it is not right-continuous at 0. A distribution function satisfies F(x)=\mathrm{P}(X\le x) and therefore includes an atom in the value at the atom. Set F(x)=0 for x<0 and F(x)=1 for x\ge 0. That function is the distribution function of a point mass at 0.