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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?

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up vote 3 down vote accepted

Assume that $N(f)=0$ for $f\colon [0,1]\to \Bbb R$ a continuous function. We get $xf(x)=0$ for all $x\in [0,1]$. In particular, for $x\neq 0$, this gives $f(x)=0$. Now, we have by continuity $$f(0)=\lim_{n\to +\infty}f(n^{-1})=0$$ as $f(n^{-1})=0$ for each integer $n$.

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