  We show that for long-range percolation with polynomially decaying connection
probabilities in dimension $d\geq 2$, the critical value depends continuously
on the precise specifications of the model. Among other things, we use this
result to show transience of the infinite supercritical long-range percolation
cluster in dimension $d\geq 3$ and to prove a shape theorem for super-critical
long-range percolation in the strong decay regime.
