  Bent functions can be classified into regular bent functions, weakly regular
but not regular bent functions, and non-weakly regular bent functions. Regular
and weakly regular bent functions always appear in pairs since their duals are
also bent functions. In general this does not apply to non-weaky regular bent
functions. However, the first known construction of non-weakly regular bent
functions by Ce\c{s}melio\u{g}lu et {\it al.}, 2012, yields bent functions for
which the dual is also bent. In this paper the first construction of non-weakly
regular bent functions for which the dual is not bent is presented. We call
such functions non-dual-bent functions. Until now, only sporadic examples found
via computer search were known. We then show that with the direct sum of bent
functions and with the construction by Ce\c{s}melio\u{g}lu et {\it al.} one can
obtain infinitely many non-dual-bent functions once one example of a
non-dual-bent function is known.
