Space of continuous functions form a Banach space?

Question

Let $X$ be the collection of all continuous real-valued functions defined by \begin{equation*} ||f||=\sup_{x\neq y}\frac{\left\vert f(x)-f(y)\right\vert }{\left\vert x-y\right\vert } \end{equation*} for all $f \in X$, then $(X, ||\cdot||)$ is a Banach space.

Answer 1

No, you can't. There are $2$ problems:

• $f(0)=0$ guarantees that $\lVert . \rVert$ is a norm.
• It won't even be a vector space, because if we take a function $f$ with $f(0)=c>0$, and pick a scalar $\alpha < 0$, we get that $\alpha f(0) = \alpha c < 0$, which can't be in the set.