  This paper contains a number of observations on the {$F$-signature} of
triples $(R,\Delta,\ba^t)$ introduced in our previous joint work. We first show
that the $F$-signature $s(R,\Delta,\ba^t)$ is continuous as a function of $t$,
and for principal ideals $\ba$ even convex. We then further deduce, for fixed
$t$, that the $F$-signature is lower semi-continuous as a function on $\Spec R$
when $R$ is regular and $\ba$ is principal. We also point out the close
relationship of the signature function in this setting to the works of Monsky
and Teixeira on Hilbert-Kunz multiplicity and $p$-fractals. Finally, we
conclude by showing that the minimal log discrepancy of an arbitrary triple
$(R,\Delta,\ba^t)$ is an upper bound for the $F$-signature.
