The question is
$F$ is a distribution function. Show that $$P\{X = x\} = F(x) - \lim_{y \uparrow x}F(y)$$
This is trivial. But how can we prove it in a formal way?
The question is
$F$ is a distribution function. Show that $$P\{X = x\} = F(x) - \lim_{y \uparrow x}F(y)$$
This is trivial. But how can we prove it in a formal way?
$F(y) = \Pr (X \le y)$ so $$\lim_{y \uparrow x}F(y) =\lim_{y \uparrow x} \Pr (X \le y) = \Pr (X \lt x) $$ and thus $$F(x) - \lim_{y \uparrow x}F(y)= \Pr (X \le x)- \Pr (X \lt x)= \Pr (X = x).$$