I'm trying to show that, given X is the set of all continuous functions $\displaystyle f:[a,b] \rightarrow R $, and d(f,g) = $\displaystyle \int_{a}^{b} | f(t) - g(t) | dt $, that (X, d) defines a metric space.

My confusion is coming in when I let a = 0, b = $\displaystyle \frac{\pi}{2} $, and f = sine and g = cosine. Then wouldn't the distance function between these two be

$\displaystyle \int_{0}^{ \frac{\pi}{2}} | sin(t) - cos(t) | dt $

$\displaystyle = - (cos(\frac{\pi}{2}) - cos(0)) - ( sin(\frac{\pi}{2}) - sin(0)) $

$\displaystyle = -(0 - 1) - (1 - 0) = 1 - 1 = 0 $

But f != g, so isn't that a problem?