Let $C([-1,1]$ be the set of continuous functions on the interval $[-1,1]$, and let $S$ be the subset of $C([-1,1])$ consisting of $f$ such that $f(-x)=-f(x)$ for any $x$ in $[-1,1]$. Is $S$ a vector space?

So this is one of the questions from my textbook and from the answers in the textbook, I know that $S$ is a vector space. I am having trouble with determining of how $S$ is a vector space when it involves intervals.

I know the axioms for determining whether something is a vector space but how would I apply them with the intervals? 

Any help would be appreciated. Thanks.