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The object of this article is a family of $q$-series originating fromHabiro's work on the Witten-Reshetikhin-Turaev invariants. The $q$-seriesusually make sense only when $q$ is a root of unity, but for some instances, italso determines a holomorphic function on the open unit disc. Such an exampleis Habiro's unified WRT invariant $H(q)$ for the Poincar\'{e} homology sphere.In 2007, Hikami observed its discontinuity at roots of unity. More precisely,the value of $H(\zeta)$ at a root of unity is $1/2$ times the limit value of$H(q)$ as $q$ tends towards $\zeta$ radially within the unit disc. In thisarticle, we explain the appearance of the $1/2$-factor and generalize Hikami'sobservations by using Bailey's lemma and the theory of false theta functions.

Hikami's observations on unified WRT invariants and false theta functions