Consider $C[0,1]$, the vector space of continuous functions on the interval $[0,1]$ and consider the map $N:[0,1]\to\mathbb R$ where $N(f)=\sup_{x\in[0,1]}x\cdot|f(x)|.$
I was able to show the homogeneity and subadditivity of $N$, but I do not know how to prove that $f=0\iff N(f)=0$. I feel like this follows from continuity, but I do not know how to give a proper $\epsilon$-$\delta$ proof. Can someone help me out?