I would like to ask, if applying makeup outside your home for any function is haram?
Because we have these statements that if a husband allows you to wear makeup and go out without hijaab then t is allowed and there is no sin to it. How true is this?

Applying makeup is itself not Haram but if after makeup anyone wants to go outside in the presence of non mahram then she has to hide her makeup. Otherwise if she is going to the place where there is no non mahram then its ok. A woman can not go outside without a proper covering (Hijab) even if her husband allows her. It is haram for a woman to go outside without hijab even if her Father or Brother or mother or even her husband ask her to do so.