  We previously investigated the compatibility of the
$Kimber$-$Martin$-$Ryskin$ ($KMR$) $unintegrated$ parton distribution functions
($UPDF$) with the experimental data on the proton (longitudinal) structure
functions ($PSF$ ($PLSF$)). Recently $Golec-Biernat$ and $Stasto$ ($GBS$)
demonstrated that the differential version of $KMR$ prescription and the
implementations of angular (strong) ordering ($AOC$ ($SOC$)) constraints, cause
the negative-discontinuous $UPDF$ with the ordinary parton distribution
functions ($PDF$) as the input, which leads to a sizable effect on the
calculation of $PSF$. In the present work, we use the new $MMHT2014$-$LO$-$PDF$
as the input and focus on the $UPDF$ behaviors as was raised by $GBS$.
  The resulting $PSF$ and $PLSF$ are compared with the $MSTW2008$-$LO$-$PDF$
and $MRST99$-$PDF$ and the 2014 data given by the $ZEUS$ and $H1$
collaborations. The calculated $PSF$ and $PLSF$ based on the integral
prescription of the $KMR$-$UPDF$ with the $AOC$ and the ordinary $PDF$ as the
input are reasonably consistent with the experimental data. Therefore, they are
approximately independent to the $PDF$ (no need to impose cutoff on the $PDF$).
  At very small $x$ regions because of the excess of gluons in the
$MMHT2014$-$LO$-$PDF$ and $MSTW2008$-$LO$-$PDF$, an increase in $PSF$ and
$PLSF$ is achieved. Finally, according to the $GBS$ report the differential
version by using the cutoff independent $PDF$ produces results far from the
experimental data.
