  We consider two different objects on super-critical Bernoulli percolation on
$\mathbb{Z}^d$ : the time constant for i.i.d. first-passage percolation (for
$d\geq 2$) and the isoperimetric constant (for $d=2$). We prove that both
objects are continuous with respect to the law of the environment. More
precisely we prove that the isoperimetric constant of supercritical percolation
in $\mathbb{Z}^2$ is continuous in the percolation parameter. As a corollary we
prove that normalized sets achieving the isoperimetric constant are continuous
with respect to the Hausdroff metric. Concerning first-passage percolation,
equivalently we consider the model of i.i.d. first-passage percolation on
$\mathbb{Z}^d$ with possibly infinite passage times: we associate with each
edge $e$ of the graph a passage time $t(e)$ taking values in $[0,+\infty]$,
such that $\mathbf{P}[t(e)<+\infty] >p_c(d)$. We prove the continuity of the
time constant with respect to the law of the passage times. This extends the
continuity property previously proved by Cox and Kesten for first passage
percolation with finite passage times.
