Let $f$ be a continuous linear functional in a locally convex space $X$ and let $z\in X$ so that $f(z)=\alpha$. Let $V_z$ be an open neighborhood around $z$.
It is my intuition that $V_z$ should contain points on both sides of the hyperplane, but I don't know exactly how to prove this or even if I am correct. That is, I want to show that there exists $x,y\in V_z$ so that $f(x)<\alpha$ and $f(y)>\alpha$.
Is my intuition correct, if so, is there a source that someone could point me to or a quick argument as to why this is the case?
Suppose wlog $z = 0$. If $f$ isn't zero, there is some $x\in V_0$ (wlog symmetric) with $f(x)\ne 0$. Take now $y = -x$.