# Function $F$ that $F(x)+F(-x)=\lim_{y\rightarrow\infty}F(y), \forall x \in \mathbb{R}$

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!

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 Yes, the notation $g(x^+)$ is routinely used as a shorthand for the limit of $g(y)$ when $y\to x$ restricted to $y>x$. Likewise for $g(y^-)$ and $y ## 1 Answer If we assume that the limit on the right hand side is zero, then$F$is called antisymmetric or odd, that is, it satisfies$F(x) = -F(-x)$. Now let's assume that the limit$c = \lim_{x \to \infty} F(x) + F(-x)$exists. Then$G(x) = F(x) - \frac{c}{2}$satisfies$G(x) + G(-x) = F(x) + F(-x) - c$, hence$\lim_{y \to \infty} G(y) + G(-y) = 0$, so$G$is odd by the previous reasoning, hence$F(x) = G(x) + c/2$is the sum of an odd function and the scalar$c/2\$. Odd functions arise quite frequently and a bunch of their properties can be found on the Wikipedia page linked to above.

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 Thanks! Sorry about some confusion and I just made some change. – Tim May 15 '11 at 18:08 @Tim: Might want to change the title too. – Ben Alpert May 15 '11 at 18:09 @Theo: I think you should add that if the limit is non-zero then it is an odd function + a scalar. – Asaf Karagila May 15 '11 at 18:14 @Asaf: Thanks, done. – t.b. May 15 '11 at 18:22 @Theo: Thanks! Nice to know its relation to odd function. – Tim May 15 '11 at 18:41