How we change what others think, feel, believe and do
When you make a mistake, it is often best to confidently admit it rather than try to cover it up.
With minor mistakes such as slips of the tongue, do not disturb the flow by apologizing -- just continue.
If you make a simple mistake cheerfully admit it, especially if it is not central to your premise or you have not clearly erred. Do not over-dramatize errors with abject apology.
If somebody in your audience points out an error, always treat this seriously and never dismiss them as foolish in any way. Pause to reflect and consider whether it they have a valid point. If so, nod and say something like 'I believe you have a good point there. I'm going to have to think about that.'
It is often a good idea to bounce a challenge back either to the questioner or the whole audience, asking them either a simple question of what they would do about the problem as identified. Beware in this of disappearing down a rabbit-warren of analysis and discussion, curtailing discussion as needed.
If you are not sure if something is wrong, consider whether it is worth disrupting the flow and confusing your audience or stopping. If it is pointed out to you, then acknowledge the possibility of error and say you will investigate.
I am presenting on particle physics when a member of the audience challenges a point I make. I pause and say 'I think you may have a point there -- do you have a suggestion as to an alternative?' They make a suggestion. I thank them and move on.
When something appears wrong, most speakers feel a tension between owning up and bluffing it through. Many try the latter approach as they fear ridicule and rejection over being less expert than they want to appear. However, if the audience believes you are wrong and that you are also trying to cover up, then you will get rejected anyway and also thought of with disgust, which is the most powerful rejection emotion.
On the other hand, if you handle small mistakes with humor and larger mistakes with due seriousness, then you will be seen as a worthy person and your reputation will most likely be enhanced.
Handling mistakes starts with how you think about yourself. If you think you have to be perfect or admire yourself too much, then you will likely find admitting mistakes rather difficult. On the other hand if you are comfortable in your own skin, can laugh at yourself and have a strong sense of integrity, then you will be able to handle mistakes with aplomb.