The set $C(\mathbb{R})$ of bounded continuous functions on $\mathbb{R}$ is a Banach space when equipped with the sup norm. In my understanding, it just follows from the fact that a Cauchy sequence of continuous functions converges uniformly to a continuous function.
How about the set, which I will denote $C_{\rm{d}}(\mathbb{R})$, of bounded functions, which are continuous except at $x=0$, where a jump discontinuity is allowed, i.e. $\displaystyle{\lim_{x\rightarrow 0^\pm}f(x)=f^\pm}$ both exist. Is that a Banach space under the sup norm?
It seems like yes, since I can apply the Cauchy argument to both intervals $(-\infty,0]$ and $[0,\infty)$ and conclude that a Cauchy sequence on $C_{\rm{d}}(\mathbb{R})$ will uniformly limit to a function, which is continuous on both intervals individually.
Am I right or completely wrong? If I am right, is there a name for such a space?
Best Answer
You're correct. Try showing that $C_d(\Bbb R)$ is isomorphic(as a normed linear space) to $C((-\infty,0]) \oplus C([0,\infty))$. Since the spaces $C((-\infty,0])$ and $C([0,\infty))$ are individually Banach spaces, so is their direct sum. As far as I know, there is no special name for this space.