Does the CMF of the degenerate distribution have to be right continuous?

Question

The CMF of the degenerate distribution at 0 is defined on Wikipedia as below:

$$ F(x)= \begin{cases} 0 & \space \text{for} \space x<0 \\ 1 & \space \text{for} \space x\ge 0 \end{cases} $$

My question is, why do we define $F(0)=1$? Is it allowed define $F(0)=0$ and thus making $F$ left continuous?

score 1 · Accepted Answer · answered Jun 27 '19 at 11:50

If you define it as $F(x)=\mathbb{P}(X<x)$ then it is automatically left continuous. We don't define it to be left or right continuous.

If you define it as $F(x)=\mathbb{P}(X\leq x)$, then it is right continuous. This follows from continuity of measures.

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