Question

\[d(f,g)=max_{0\leq t\leq 1}|f(t)-g(t)| \]define a metric on
C[a,b]
i have proved yes. can continuous functions on [0,1] be replaced by "bounded"? i know a continuous function takes on its maximum, but a bounded one does not.

Answer 1

must i replace "max" by "sup"?

Answer 2

Yes. For example:
f(x) = x , x in [0.1)
= 0 , x = 1

f is not continuous, it is bounded, but max |f| on [0.1] is not defined. However if it's bounded then sup |f| is always defined.

Answer 3

thank you.