It would be nice to think that people are rational and can be persuaded to change their minds when presented with evidence. In reality, we're often irrational and cling to our prejudices despite voluminous information to the contrary — and when that happens, the most sensible strategy is just to walk away.
Urban legend site Snopes provides a neat demonstration of this principle with an account of how they received numerous emails accusing them of bias and inaccurate research after they debunked a news story relating to the appointment of a judge by US President Barack Obama. In every case, it was evident that the complainants had not actually read the article in question: they'd just taken at face value claims by others that the research was biased and inaccurate. As Snopes concludes:
A good many people are determined to believe whatever they want to believe, and no collection of contradictory factual information, no matter how large or authoritative or impressive it might be, is ever going to dissuade them from their beliefs.
One of my New Year's resolutions is to remember this, and to spend less time trying to persuade people that they've got the wrong end of the stick. If they're determined to be ignorant, it's wasting everybody's time to try and change their minds.