  The model of rigid linear heat conductor with memory is reconsidered
focussing the interest on the heat relaxation function. Thus, the definitions
of heat flux and thermal work are revised to understand where changes are
required when the heat flux relaxation function $k$ is assumed to be unbounded
at the initial time $t=0$. That is, it is represented by a regular integrable
function, namely $k\in L^1(\R^+)$, but its time derivative is not integrable,
that is $\dot k\notin L^1(\R^+)$. Notably, also under these relaxed assumptions
on $k$, whenever the heat flux is the same also the related thermal work is the
same. Thus, also in the case under investigation, the notion of equivalence is
introduced and its physical relevance is pointed out.
