Continuing is a game mechanic featured in many video games, Mario games being only some of them. When a player gets a Game Over, they will sometimes be given the option to continue, which restarts them in a certain place (usually the beginning of the world they were in) with a whole new set of lives.
In games with the option of saving, there is no Continue when not enough amounts of what is required, but the player can restart from where they last saved. This is similar to the Continue.
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The Continue screen as seen during All-Star mode in Super Smash Bros. Brawl.