  We use a fully self-consistent Hartree$-$Fock (HF) based continuum random
phase approximation (CRPA) to calculate strength functions $S(E)$ and
transition densities $\rho_t(r)$ for isoscalar giant resonances with
multipolarities $L = 0$, 1 and 2 in $^{80}Zr$ nucleus. In particular, we
consider the effects of spurious state mixing (SSM) in the isoscalar giant
dipole resonance (ISGDR) and extend the projection method to determine the
mixing amplitude of spurious state so that properly normalized $S(E)$ and
$\rho_t(r)$ having no contribution due to SSM can be obtained. For the
calculation to be highly accurate we use a very fine radial mesh (0.04 fm) and
zero smearing width in HF$-$CRPA calculations. We first use our most accurate
results as a basis to establish the credibility of the projection method,
employed to eliminate the SSM, and then to investigate the consequences of the
common violation of self-consistency, in actual implementation of HF based CRPA
and discretized RPA (DRPA), as often encountered in the published literature.
The HF$-$DRPA calculations are carried out using a typical box size of 12 fm
and a very large box of 72 fm, for different values of particle-hole energy
cutoff ranging from 50 to 600 MeV.
