Show the following is a metric

[DotKite]

Homework Statement

Show that $d(f,g) = \int_{0}^{1}\left | f(x) - g(x) \right | dx$ is a distance function. Where $f : [0,1] \rightarrow R$ and $f$ is continuous.

Homework Equations

The Attempt at a Solution

I am stuck on the second property where you have to show d(f,g) = 0 iff f = g. THe left direction is trivial. However d(f,g) = 0 implying f=g is giving me trouble. I have tried contrapositive, but it doesn't seem to be getting me anywhere.

 

[Dick]

Homework Statement

Show that $d(f,g) = \int_{0}^{1}\left | f(x) - g(x) \right | dx$ is a distance function. Where $f : [0,1] \rightarrow R$ and $f$ is continuous.

Homework Equations

The Attempt at a Solution

I am stuck on the second property where you have to show d(f,g) = 0 iff f = g. THe left direction is trivial. However d(f,g) = 0 implying f=g is giving me trouble. I have tried contrapositive, but it doesn't seem to be getting me anywhere.

The fact you need is that if F(x)>=0 on [0,1], F(a)>0 for some $a$ in [0,1] and F(x) is continuous (very important) then $\int_{0}^{1}\left | F(x) \right | dx \gt 0$. Can you figure out how to prove that?

 

[DotKite]

The fact you need is that if F(x)>=0 on [0,1], F(a)>0 for some $a$ in [0,1] and F(x) is continuous (very important) then $\int_{0}^{1}\left | F(x) \right | dx \ge 0$. Can you figure out how to prove that?

wouldn't I need to show that $\int_{0}^{1}\left | F(x) \right | dx > 0$
to get the contrapositive?

 

[DotKite]

Oh I see! ok. NVM. Gonna try to prove it

 

[Dick]

wouldn't I need to show that $\int_{0}^{1}\left | F(x) \right | dx > 0$
to get the contrapositive?

Yes, of course. Typo. Sorry. I corrected it.