Subspace in C[0,1]

Question

Question: Let W = set of all functions which vanishes at some point in interval (0,1). Is W a subspace of C[0,1].

Since (af+bg)(x) where a,b is in R will also vanish at some point in (0,1) this will form a subspace.

Am I right?

Answer 1

If they do not need to vanish at the same point, then this is not a linear subspace: $f(x) = x-\frac14$ vanished at $\frac14$, $g(x)= \frac34-x$ at $x=\frac34$, but $f+g= \frac12$ for all $x$ and this vanishes nowhere.