Homework Help: Why d is not a metric

1. May 11, 2009

happybear

1. The problem statement, all variables and given/known data

If d= sup{|f(x)-g(x)|} where f, g are contuous function from R to R, why is d not a metric

2. Relevant equations

3. The attempt at a solution

2. May 11, 2009

quasar987

Take f(x)=x, g(x)=0. then d(f,g)=infinity. So d is not a map into R.

3. May 11, 2009

happybear

why infinity is not a map into R?

4. May 11, 2009

quasar987

I said that d is not a map into R. And this is because if you take f(x)=x and g(x)=0, then d(f,g)=sup{|x|:x in R} is not an element of R. Indeed, the real number sup{|x|:x in R}, if it exists, is an upper bound for the set {|x|:x in R}. But this set is unbounded from above and hence possesses no upper bound so sup{|f(x)|:x in R} does not exists.