If you have a CDF that's say:
F(x) = 1-e^(-.43t) for t ≥ 0
0 t < 0
And you were trying to prove that it was continuous, the book/lectures say that you only have to prove that something is continuous from the right:
So you'd use:
Lim F(x) = F(0)
t-->0 (+, from the right)
But what I don't get is how this can prove continuity at all. You've basically annexed the bottom part of the equation (0 t<0) from the entire story. All you're really saying is what's already written there, that you equal F(0) as you approach from the right. BUT since this function put a greater than or EQUAL to sign on the restrictions of the actual rightside function, you're not even touching the left function.
This especially confuses me in that if you want to show continuity then, in a lot of these functions, where you get free range on what to make just greater than or what to make greater than or equal to, you could potentially get the same function be continuous or not continuous depending on where you put the "or equal to" sign.
I asked Prof. Greene about this, but I don't think I was able to properly convey what I was asking. Nonetheless she told me to take the limit of the bottom function as it approaches c from the left, and then take the limit of the right function as it approaches c from the right and make sure that those are equal. I.e.:
Lim F(x) = Lim F(x)
t-->0 (+, from the right) t-->0 (-, from the left)
But I feel like that doesn't really fit with what is the bare minimum of proving a CDF which is just that it is continuous from the right. I.e.:
Lim F(x) = F(0)
t-->0 (+, from the right)
What if I get a function out there that's only continuous from the right and not continuous from the left, and try to make sure that the limit approaching from the right and the limit approaching from the left are equal, I would be excluding a whole bunch of functions that are only right continuous. Furthermore, even if you did make sure that the limits were equal as you approached from the right or left, you could have a jump discontinuity that would still make the two functions discontinuous..
Something does appear strange to me here. In general, as your Prof. Greene said, if the limit of the function is the same from the left and right at the point in question (
AND the function is defined there), the function is continuous. That is, if they are the same from both the left and right they can't jump.
In some situations, this may entail only needing to show the one sided limit, such as the function
\(\displaystyle F(x)\, =\, \begin{cases}1\, -\, e^{-43\,t} & \mbox{for }\, 0\, \leq\, t \\ 0 & \mbox{for }t\, <\, 0 \end{cases}\)
We know that the function is continuous at zero from the
right so that we only need to show it is continuous from the
left. However, that is not generally the case. A more formal definition is given, for example, at:
http://mathworld.wolfram.com/ContinuousFunction.html
but notice that a function is continuous at c iff it is continuous from both the left and right at c.
