How to prove that whether it is a Banach space or not? We consider the Banach space of all continuous functions on $X$ such that for each $f$ in the space, 
\begin{equation*}
||f||=\sup_{x\neq y}\frac{\left\vert
f(x)-f(y)\right\vert }{\left\vert x-y\right\vert }.
\end{equation*}
How I can prove that it is a Banach space? 

$m$ is not a norm on the set of the continuous function on $[0, 1]$ with $f(0)=0$, because not all continuous functions on $[0, 1]$ are Lipschitz. For example, take $f(x)=\sqrt{x}$. It's continuous, $f(0)=0$, but
$$\frac{|\sqrt{x}-\sqrt{y}|}{|x-y|}=\frac{1}{|\sqrt{x}+\sqrt{y}|}$$
And taking $x=0$, we get that
$$m(f)\geqslant \lim_{y \to 0+0} \frac{1}{\sqrt{y}}=+\infty$$
