TITLE: Is this space Dense in $ C[0,1]$
QUESTION [5 upvotes]: Is the space $\{f\in C[0,1]\mid \int_0^1f\neq 0\}$ dense in $C[0,1]$ with sup-norm topology. 
I think yes, because it is the inverse image of the set $\mathbb{R} \setminus \{0\}$

REPLY [6 votes]: Call that set $A$, we shall prove it is a dense set.
Pick $f\not \in A$, in other words $f$ such that $\int\limits_{0}^1 f=0$, and pick $\epsilon >0$.
Notice that the function $g(x)=f(x)+\epsilon/2$ is in $A$ and its distance from $f$ is $\epsilon/2$, so the set is in fact dense.