Pointwise addition
= Pointwise addition
{title2=$(f+g)(x)=f(x)+g(x)$}
For <functions> $f,g:X\to V$ taking values in the same <vector space>, <pointwise addition> is defined by $(f+g)(x)=f(x)+g(x)$ for every $x\in X$. For $f(x)=x$ and $g(x)=x^2$, the resulting <function> is $x\mapsto x+x^2$; at $x=2$ its value is six.
The <zero function> and pointwise <scalar multiplication> make the collection of all such <functions> a <vector space>. The <vector space> identities follow by evaluating at an arbitrary input and using the corresponding identities in $V$. This construction also preserves <continuity> and <linearity> when the original <functions> have those properties, and is the addition used in the <superposition principle> for a <linear differential equation>.