You need to save to keep you in-game progress. What's more, it helps to so often—lest you have to redo all your hard work. If you were wondering if there is a saving feature in Lost in Random, the answer is yes.
However, there is no way to manually save in Lost in Random. Going into the menu only gives you the options of Continue, Options, or Quit to the main menu. There is no save feature in your inventory, either.
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What the game does feature is an autosave system. The issue with that is there is no notification on the screen at all to tell you when the game is saving. This means it is impossible to tell when Lost in Random is saving your progress. It seems that the game may autosave when you complete an objective, but there is no way to tell for sure.
When you try to exit the game, there is no prompt that asks you if you are sure you want to quit without saving, neither is there a prompt that notifies you that you may lose your progress. This makes it risky to quit out of the game unless you have completed an objective or reached an obvious checkpoint. Fortunately, access to three save files provides room to experiment without messing up your main game.
For more information on Lost in Random, check out Where to find the Storybook Pages in Onecroft in Lost in Random and Is there a map in Lost in Random? on Pro Game Guides.