Q:

Check vector space

From these which isn't subspace of $C(a,b)$:
a) $\{f\in C(a,b): \lim_{x\to(a+b)/2} f(x)=0\}$
b) $\{f\in C(a,b): \lim_{x\to (a+b)/2} f(x)=1\}$
c) $\{f\in C(-\infty,+\infty): f(x)=f(-x)\}$
d) all of these 
Can anyone explain!

A:

What do you know about the axioms of a subspace you have to check? 
 For instance, every subspace of a vector space has to contain the zero vector:
 Concerning (b): $ M :=\{f∈C((a,b)):\lim_{x→(a+b)/2}f(x)=1\}$ cant be a subspace of $C((a,b))$, since the function $f \equiv 0 \notin M$. I gave that as an example. Try the others by checking the axioms: Let $V$ be a vector space over a field $K$. $W$ is a subspace of $V$, if all of the following axioms hold:

The zero vector is in  $W$.
If $u$ and $v$ are elements of $W$, then their sum $u$ + $v$ is an element of
$W$.
If v is an element of W and $\lambda $ is an element from $K$, then
their product  $\lambda v$ is an element of $W$.

