DearMHFmembers,

I have two questions on Banach spaces,

and I would really appreciate if you help me with acompleteanswer.

Let $\displaystyle [a,b]$ be a interval and $\displaystyle c\in[a,b]$.

- Denote by $\displaystyle \Omega_{c}([a,b],\mathbb{R})$ the set of all real valued bounded functions on $\displaystyle [a,b]$,

whichmayhavejump type discontinuityfrom the left-hand side at the point $\displaystyle c$,

i.e., $\displaystyle \lim_{t\to c^{-}}f(t)\neq f(c)$ and $\displaystyle \lim_{t\to c^{+}}f(t)=f(c)$.

Is $\displaystyle \Omega_{c}([a,b],\mathbb{R})$ a Banach space with the $\displaystyle \sup$ norm?- Denote by $\displaystyle \Psi_{c}([a,b],\mathbb{R})$ the set of all real valued bounded functions on $\displaystyle [a,b]$,

whichmayhaveremovable type discontinuityat the point $\displaystyle c$.

Is $\displaystyle \Psi_{c}([a,b],\mathbb{R})$ a Banach space with the $\displaystyle \sup$ norm?

Thanks.

bkarpuz