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?