Metric spaces

[michelle10]

I have to show that, for the set of all continous functions on [0,1], the distance

is a metric.

I don't know how to prove that if d(f,g)=0 then f=g.

Please, help me!
Thanks.

 

[Steven G]

I have to show that, for the set of all continous functions on [0,1], the distance
View attachment 4995
is a metric.

I don't know how to prove that if d(f,g)=0 then f=g.

Please, help me!
Thanks.

If h(x) is never negative then every definite integral in the form of int(h(x)dx) will never be negative. Do you see that? The definite integral is the area between the curve and the x-axis. If the function (or part of it) is under the x-axis then the area between the function and the axis will be negative.
So if h(x) is non negative on (a,b) and integral of h(x)dx from a to b is zero then what can you say about h(x)?? You really should see this way before playing with metric spaces.