Checking Cumulative distribution function Two functions are given as below. How to check whether the functions are cdf or not?
$$F(x) = \begin{cases}
0 ; & x<0 \\ 
x ; & 0\leqslant x<\frac{1}{2}\\ 
1 ; & x\geq \frac{1}{2}
\end{cases}$$
and 
$$F(x) = \begin{cases}
0 ; & x<0 \\ 
x ; & 0\leqslant x \leqslant\frac{1}{2}\\ 
1 ; & x> \frac{1}{2}
\end{cases}$$
 A: In the US and western Europe it is customary to define CDFs as right
continuous. In some parts of eastern Europe and Asia it has been customary to
define CDFs as left continuous. [See for example, Gnedenko (tr. from Russian by Seckler) Theory of Probability, Chelsea, 1967. "Let $\xi$ be a random
variable and $x$ an arbitrary real number, ... $F(x) = P(\xi < x).$"] 
So check the definition in your text; the "correct" answer may depend on
where you are. 
As I read the definitions of the two $F$s in your Question, their respective plots are
as follows. Blue dots are determined by equal signs in the definitions.

If the value of the function in the 'middle part' were $2x,$ then
curve would be continuous provided it is defined at all points (including $x=0$ and $1/2$).

Notes: Other things to check in determining whether a curve is a CDF are
(a) whether it is nondecreasing, (b) whether it becomes or tends to 0
for sufficiently small $x$, and (c) whether it becomes or tends to 1 for
sufficiently large $x.$
