  We explore a variant of the zeta function interpolating the Bernoulli numbers
based on an integral representation suggested by J. Jensen. The Bernoulli
function $\operatorname{B}(s, v) = - s\, \zeta(1-s, v)$ can be introduced
independently of the zeta function if it is based on a formula first given by
Jensen in 1895. We examine the functional equation of $\operatorname{B}(s, v)$
and its representation by the Riemann $\zeta$ and $\xi$ function, and recast
classical results of Hadamard, Worpitzky, and Hasse in terms of
$\operatorname{B}(s, v).$ The extended Bernoulli function defines the Bernoulli
numbers for odd indices basing them on rational numbers studied by Euler in
1735 that underlie the Euler and Andr\'{e} numbers. The Euler function is
introduced as the difference between values of the Hurwitz-Bernoulli function.
The Andr\'{e} function and the Seki function are the unsigned versions of the
extended Euler resp. Bernoulli function.
