Tricky question ... as the concept of rapier is a bit discussable.
... Chris Evans would better guide you into that universe
.
Speaking of a "true" rapier the blade would be completely dull.
I have once seen a rapier blade, which blade was a highly tense diamond section dull "iron". I believe even their points don't have to be very sharp, due to their thrusting abilities.
Swords like the ones posted here are rapiers of second generation, to put it that way; their blades being more or less sharp like many other swords; not razor sharp ... simply sharp.
I guess this evolution was mostly based on the fact that original rapiers were destined for restricted purposes, to be used by previleged classes, either in fencing schools or, when in street fights, obliging for both opponents to be school trained.
... Whereas later ones could be effective in the hands of the common man, something more realistic as, when the moment comes, fighting is how you manage do it and not by following demanding manuals.
I hope all this makes some sense
