Last Thursday's presidential debate was, at times, quite rough for Donald Trump — he was subjected to difficult questioning, and his responses were frequently controversial. As a result, some pundits speculated that the Republican poll leader's support, at long last, might begin to decline — or even hemorrhage, considering how one focus group recoiled at seeing Trump in action.

Yet four new polls have come out since the debate, and all of them show Trump remaining at the front of the GOP pack.

A Reuters/Ipsos poll found Trump's support unchanged from the week prior to the debate, at 24 percent, with his nearest rival being Jeb Bush, at 12 percent. A poll from Morning Consult actually found Trump's support increasing 7 points after the debate, to 32 percent (with Bush in second, at 11 percent). And an NBC/Survey Monkey poll found Trump at 23 percent, with his closest competitor Ted Cruz, at 13 percent.

All of these polls were conducted via internet, not by phone. But poll analysts have argued that sophisticated internet polling is worth taking seriously, with SurveyMonkey's operation (run by Jon Cohen, the former head of polling at the Washington Post) coming in for particular praise:

Context for new NBC poll - Cohen is serious & quality online polls now easily rival phone on accuracy. Don't dismiss https://t.co/igZ43FC8Aq — Brendan Nyhan (@BrendanNyhan) August 9, 2015

Whether a poll is conducted online or by phone just isn't as important as it was 5 years ago. — Nate Cohn (@Nate_Cohn) August 9, 2015

You just can't judge simply off of mode anymore. Best phone probably > best internet, but best internet definitely > bad phone. — Nate Cohn (@Nate_Cohn) August 9, 2015

A fourth post-debate poll from Public Policy Polling surveyed Iowa Republicans and found Trump remaining in first place there, too. This is a phone robo-poll run by a Democratic-aligned firm, and it found Trump leading at 19 percent, and Ben Carson and Scott Walker tied for second with 12 percent each.

A good rule of thumb for following polling is not to overanalyze individual poll results — averages are a better clue as to what's going on. We're still waiting for a new live interview poll, but for now the fact that none of these four polls shows a dramatic Trump decline — and that one even shows his strength increasing — seems to suggest that there's been no such decline. Trump's support from a quarter or so of Republican voters, it appears, is more durable than many expected.

VIDEO: Donald Trump explains why he became a Republican