Determine the Values of $a$ so that $W_a$ is a Subspace

Problem 662

For what real values of $a$ is the set
\[W_a = \{ f \in C(\mathbb{R}) \mid f(0) = a \}\]
a subspace of the vector space $C(\mathbb{R})$ of all real-valued functions?

The zero element of $C(\mathbb{R})$ is the function $\mathbf{0}$ defined by $\mathbf{0}(x) = 0$. This shows that $\mathbf{0} \in W_a$ if and only if $a=0$.

We have shown that if $a \neq 0$, then $W_a$ is not a subspace as every subspace contains the zero vector. Now we consider the case $a=0$ and prove that $W_0$ is a subspace.

We verify the subspace criteria: the zero vector of $C(\R)$ is in $W_0$, and $W_0$ is closed under addition and scalar multiplication.

As mentioned before, $\mathbf{0} \in W_0$.

Now suppose $f, g \in W_0$. Then $f(0) = g(0) = 0$, and so
\[(f+g)(0) = f(0) + g(0) = 0.\]
Thus $f+g \in W_0$. Finally, if $c \in \mathbb{R}$ is a scalar and $f \in W_0$, then
\[(cf)(0) = c f(0) = c \cdot 0 = 0.\]
Thus $cf \in W_0$, and $W_0$ is a vector subspace.

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