I was wondering if there is a name for function $F: \mathbb{R} \rightarrow \mathbb{R}$ with property $F(x)+F(-x)=\lim_{y\rightarrow\infty}F(y), \forall x \in \mathbb{R}?$
An example is when $F$ is the probability distribution function of a symmetric density function. A random variable with such probability distribution function is called symmetric.
I would like to know if properties of such kind of functions have been studied both generally and especially for probability theory i.e. when $F$ is a probability distribution function.
Added: Also I wonder in Kai Lai Chung's A Course in Probability Theory where he wrote for the probability distribution function of a symmetric distribution $F(x)=1-F(-x-), \forall x \in \mathbb{R},$ what is the meaning of $-x-$? I guess that because a probability distribution function is a right-continuous function, maybe Chung wanted to emphasize that it is the left limit that is used to define symmetric distribution.
Thanks and regards!