It's very common for animated shows to have a complete disregard for continuity. A character could fall off a cliff, and be destined for a 'life-long recovery', but are completely fine by the next episode, as if it never happened. It's all part of the slapstick element that often accompanies the realm of cartoons.
However, Gravity Falls pays close attention to its continuity and makes sure that any event that occurs, no matter how insignificant, has an affect on all the future outcomes. The most well-known example within the show is the 'S' letter of the mystery shack sign being cut off in "headhunters" and staying that way for the rest of the series.
When Stan applies a bandage to his hand during the credits in "Society of the Blind Eye", he is seen wearing it in "Blendin's Game" and "The Love God". Likewise, when Mabel wins Waddles the pig at the Mystery fair, he remains a permanent supporting character for the Pines twins for the rest of the series.