real analysis - Only removable/jump discontinuities

Definitions

Consider R with the usual metric and let $f:A \to R$ , $A \subset R$ be a function.

We say that $f$ has a jump discontinuity at a point $p \in A $ iff
$\lim_{x \to p^-}f(x)$ and $\lim_{x \to p^+}f(x) $ exists but have different values.

We say $f$ has a removable discontinuity at a point $p \in A$ iff
$\lim_{x \to p}f(x)$ exists but $\lim_{x \to p}f(x) \neq f(p) $ .

Question

I'm trying to find two functions with $R$ as co-domain that are everywhere discontinuous ( discontinuous at every point of the domain ) such that one only has jump disconinuities and the other only has removable discontinuities.
Is there any pair of examples showing this is possible,or can we prove that this is not possible ?
$A$ is an arbitrary subset of $R$.

Thanks a lot in advance.