CDF - Cumulative distribution function If the CDF is defined:  $F_X\left(x\right)=\:1-e^{-λx}\::\:x\ge 0\:and\:0:\:x<0$ 
But when we substitute $x=0$ in the equation $1-e^{-λx}$ we got $0$. 
So why we can't say that:  $F_X\left(x\right)=\:1-e^{-λx}\::\:x>0\:and\:0:\:x\le 0$

This is the CDF of an exponential distribution which is a continuous distribution. As such it’s CDF is expected to be continuous so the left- and righthand limits of $F_X$ at $0$ should coincide.
As a result both of your definitions are valid and equally correct.
