I'm having a lot of trouble proving the left limit exists at any point.
My CDF is (1 - exp(-x/γ)). My professor told me that we should approach it as F(x - delta) = P(X < x), but I really don't know how to do that. At the same time, I'm also not entirely certain how to prove it as right-continuous, i.e. the right limit of (1 - exp(-x/γ)) = P(X < x) = F(x). Again, he subtracted delta from F(x). I don't know how you'd show that is meaningful.
I understand monotonicity and the left and right limits as they approach negative and positive infinity, respectively.