  We define a sequence of functions, namely tame cuts, in the Fourier algebra
$A(G)$ of a locally compact group $G$, that satisfies certain convergence and
growth conditions. This new consideration allows us to give a group admitting a
Fourier multiplier that is not completely bounded. Furthermore, we show that
the induction map $MA(\Gamma)\rightarrow MA(G)$ is not always continuous. We
also show how Liao's Property $(T_{Schur}, G, K)$ opposes tame cuts. Some
examples are provided.
