How to prove a type of functions is a subspace of the vector space of all functions. I've been working on algebra and want to know how to determine if a certain type of function is a subspace of the vector space $\mathbb{R} \to \mathbb{R}$.  So far I've been using the two properties of a subspace given in class when proving these sorts of questions, $$\forall w_1, w_2 \in W \Rightarrow w_1 + w_2 \in W$$   and $$\forall \alpha \in \mathbb{F}, w \in W \Rightarrow \alpha w \in W$$  The types of functions to show whether they are a subspace or not are:  (1) Functions with value $0$ on a specified set $S\subset \mathbb{R}$  (2) Functions with only finitely many discontinuity points (3) Functions with value $0$ outisde of a finite set   I mainly face difficulty generalizing these type of functions into some form say $f(a)=b$ as to have a $w_1, w_2$ to manipulate and prove whether or not it is a subspace using the properties of subspace.
 A: (1) Suppose $f(x)=g(x) = 0$ for all $x \in S$, then is the same true of $\lambda f$ and $f+g$?
(2) Suppose $f,g$ have only finitely many (possibly different) points of discontinuity. What about $\lambda f$ and $f+g$? Do they have finitely many points of discontinuity?
(3) Suppose $f,g$ are zero for all except finitely many (possibly different for $f,g$) points. Is the same true of $\lambda f$ and $f+g$?
A: 
if a certain type of function is a subspace

If the set of certain type of functions is a subspace.
Let $a,b\in K$ (the field).
(1) If $f$ and $g$ are $0$ in $S$. Then, for $s\in S$:
$$
\begin{align}
(af + bg)(s)  = (af)(s)+(bg)(s)=af(s)+bg(s) = a0+b0 = 0.
\end{align}
$$
(2) If $f$ has $n$ discontinuities and $g$ $m$, then $af+bg$ has at most (why isn't it equal?) $m+n$ discontinuities.
(3) If $f$ is non-vanishing only in a set $S$ and $g$ in a set $T$, $f+g$ will be non-vanishing in a subset of $S\cup T$ (in which case isn't it the whole union?), which is clearly finite. What about $af$?
