I have the following metric space: X=C[a,b], the family of continuous functions on [a,b] (in R). We equip the space with the metric $d(f,g)=max_{x\in[a,b]}|f(x)-g(x)|$. Fix a function $F$ in X such that $F(x)\gt0$ over all [a,b]. Now we are interested in the set $$S=(f\in X:|f(x)| \lt F(x))$$ I would like to show that the boundary of this set is all functions g such that $g(x)\leq F(x)$ for all x in [a,b] and such that $g(x)=F(x)$ for some x.
Let $r>0$ and let $f$ be a function satisfying the conditions mentioned above. I would like to show that $B(f,r)$ contains a point in S, i.e. there is a continuous functions g that stays within a distance of r from f such that $g(x)<F(x)$ for all x in [a,b].
How would I go about doing this. I know that there must be a function with this property. I am thinking that if $g$ always larger than $f-r$ and $-F$ and $g$ is always less than $f+r$ and less than F, then it will satisfy all the conditions. But how can I find such a g which is continuous?
Any help is appreciated.
Just use the function $g(x)=(1-\varepsilon)F(x)$ with suitable small $\varepsilon,$ say $\varepsilon=r/(2\max_{y\in[a,b]} F(y)).$