$d(f,g) = \int_a^b |f(x)-g(x)|dx,\ f,g \in X = C(I)$ the set of all continuous functions from the closed bounded interval $I=[a,b]$ to $\mathbb{R}$
I have found that $d$ is a metric on the set X. However
- Is this still a metric if $I=(0,1)$?
- Is this still a metric if $I=\mathbb{R}$
It seems to me that $d$ is not a metric for 1 as $f$ or $g$ could be unbounded as it is an open interval and therefore the integral would not exist?
And $d$ is not a metric for 2 for the same reason?