Derivative of cumulative distribution function Lets say  f is probability funktion to cdf F.
I want to compute $$lim_{ \Delta t \rightarrow 0+}\frac{ P( t<X<t +\Delta t )}{\Delta t}  $$
I should get $f(t) $. How do I get to that?

$$ \lim_{ \Delta t \rightarrow 0+}\frac{ P( t<X<t +\Delta t )}{\Delta t} = \lim_{ \Delta t \rightarrow 0+}\frac{ P(X < t + \Delta t) - P ( X \leq t )}{\Delta t} $$
$$ = \lim_{ \Delta t \rightarrow 0+}\frac{ F( t + \Delta t ) - F( t )}{\Delta t} = F'(t) = f(t) $$
