Finding probability from CDF Suppose that F is a cumulative distribution function where, 
$$F(x)=\begin{cases} 0,\text{if}\ x<0, \\ 1-e^{-x},\text{if}\ 0\le x<1, \\ c, \text{if}\ 1\le x<2,\\ 1-e^{-x}, \text{if}\   x\ge2, \end{cases}$$


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*Find all possible values of c.

*Find $P(0.5\le X \le 2.5)$ and $P(X=1)+P(X=2)$.
I think that the cdf should be right continuous and it is a continuous distribution (or is it mixed) so could this c have 2 values that are $1-e^-1$ and $1-e^-2$ but as the question is asking for discrete probabilities at X=1and2 then may be these are the two values at these two discrete points or are they zero? And the solution for$P(0.5\le X \le 2.5)$  must be $F(2.5)-F(.5)\\ =-e^-2.5+e^-.5$ please help me to understand the logic can be used to solve this question accurately.


Because $F(x)$ is non-decreasing, we must have $c\ge \lim_{x\to 1^-} (1-e^{-x})$. So $c\ge 1-e^{-1}$. 
A similar argument shows that $c\le 1-e^{-2}$. There are no other constraints on $c$. 
