Why Isn’t d a Metric for Continuous Functions in R?

[happybear]

Homework Statement

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

Homework Equations

The Attempt at a Solution

 

[quasar987]

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

 

[happybear]

why infinity is not a map into R?

 

[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.