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

There are infinitely many bent functions for which the dual is not bent