Is $d$ a metric on $X$?

Question

let $X= C^1([0,1], \mathbb{R})$ be the space of real value contnious function defined , differentiable and having a continious derivative on $[0,1]$. define a function fro $X \times X$ into $\mathbb{R}^{+}$ by

$d(f,g) = \sup_{x \in [0,1]} | f'(x) - g'(x)|$ where $f '$ stand for the derivative of $f$ .

Is $d$ a metric on $X$?

My attempt :yes, d is ametric on $X$. I take $f(x) = x^2$ and $g(x) =x$. Then all metric properties are satisfied

Is its true?

Any hints/ guidenace solution will be appreciated

thanks u

Answer 1

Concentrate on the question whether $d(f,g)=0$ implies $f=g.$

Answer 2

Another important point to make is that when you need to decide if something is a metric (or generally anything that has some "for all ..." qualifier), you need to actually consider all functions if you want to prove that is indeed true, while a single counterexample to one part is enough to prove it is not true.

That alone should make you understand that "it works for $f(x)=x^2, g(x)=x$" can never be proof for something that requires to work for all functions (with certain restrictions that leave more than those 2 functions in the considered set).