The article deals with functions \(\mathbb R\to\mathbb R\) that are both continuous and discontinuous on dense subsets of \(\mathbb R\), and reveal moreover some differentiability properties. The central point is a study of analytic properties of the function \(L_if\colon(0,\infty)\to\mathbb R\) of the form \[L_if(t)=\mathbb Ef\biggl(\biggl\lceil\frac{N_t+i}t\biggr\rceil\biggr),\] where \((f(k))_{k=0}^\infty\) is a sequence of reals such that \(f(k)=O(e^{\alpha k})\) for some \(\alpha>0\), \(i\in\mathbb N\cup\{0\}\), and the random variable \(N_t\) has Poisson distribution with parameter \(t\), i.e., \(N_t=k\) has probability \(t^ke^{-t}/k!\). It is proved, among others, that \(L_if\) is continuous at all irrationals and right continuous at every rational. Moreover, \(L_0f\) is right differentiable at every rational and upper left Dini differentiable at every irrational (Theorem 2.1). A certain choice of \(f\) leads to a bounded function \(\mathbb R\to\mathbb R\) that is continuous and upper left Dini differentiable on \(\mathbb R\setminus \mathbb Q\), left discontinuous and right differentiable on \(\mathbb Q\), and with a local minimum at every rational (Corollary 3.1).
The remaining part contains some results revealing the structure of an infinite \(\mathfrak c\)-dimensional space in \(\mathcal F\cup\{0\}\), for the class \(\mathcal F\) of all functions \(\mathbb R\to\mathbb R\) that are bilaterally discontinuous on a countable dense set, continuous on the complement of this set, and almost everywhere differentiable (Theorem 3.4). The result is then extended to a \(\mathfrak c\)-dimensional algebra found in \(\mathcal F\cup\{0\}\). This part goes along expected lines, with constructions based on the classical (Riemann) function \(f_\nu\): \(f_\nu(p/q)=1/q^\nu\), \(p,q\in\mathbb Z\), \(q\ne0\), \(\gcd\mkern1.4mu(p,q)=1\); \(f=0\) on \(\mathbb R\setminus \mathbb Q\).