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The Question: Assume that we are dealing with the set of all continuous functions on $[a,b]$. What can we say about $\int_{a}^b |f(x)-g(x)|dx=0$ in terms of $f(x)$ and $g(x)$.

My Question: I am trying to show that the set of all continuous functions on $[a,b]$ defines a metric with respect to the distance function defined by

$$d(f,g)=\int_{a}^b |f(x)-g(x)|dx$$

My problem is that I don't know what to say when trying to show $d(f,g)=0$ implying $f=g$. Any help on this would be appreciated.

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$$\int_a^b \vert h(x) \vert = 0 \implies h(x) = 0 \text{ almost everywhere in the interval }(a,b) \text{ (Why?)}$$ $$\text{If }h(x) = 0 \text{ almost everywhere in the interval }(a,b) \text{ and }h(x) \text{ is continuous, what can you say about }h(x)?$$ –  user17762 Jan 20 '13 at 20:26

1 Answer 1

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It's enough to show that if $h$ is a nonnegative continuous function so that if $\int_a^b h(x) dx = 0$, then $h(x) = 0$ everywhere on $[a,b]$. Assume to the contrary that there is a nonnegative continuous $h$ with $\int_a^b h(x) dx = 0$ but $h(x) = c > 0$ at some point $x \in [a,b]$. Use continuity to contradict that the integral is $0$.

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