# Math Help - Metric of C[0,1]

1. ## Metric of C[0,1]

Consider the set of $C[0,1]$, the set of all continuous functions on [0,1], and define $p(f,g)= \int ^1 _0 \mid f-g \mid dx$.

Prove that if $p(f,g)=0$, then f = g.

If $p(f,g)=0$, then I have $\int ^1 _0 \mid f(x)-g(x) \mid dx =0$.

But now, what can I do with the absolute values? Thanks!

2. Originally Posted by tttcomrader
But now, what can I do with the absolute values?
Use this property: If $h\in C[0,1]$ is a non-negative function such that $\int_0^1h(x)\,\mathrm{d}x = 0$ then $h=0$.