1. A system for improving brain function in order to prevent dementia, the system comprising: a user interface input unit to input a user's information to register a user; a user interface output unit to output information transmitted to the user from the system, or information input by the user; a user information storage unit in which the user's information input that is inputted via the user interface input unit is stored, the user information storage unit that assigns unique identification text to the user, and associate diagnosed achievement information with each identification text to thereby store associated information, the user information storage unit including identification information of the system; a brain-function-improving content storage unit, executed by a computer or a mobile terminal, that subdivides content for judging a degree of activation of human brain function into multiple classified regions and levels and stores the subdivided content; a content selection unit executed by the computer or the mobile terminal, the content selection unit that selects a content from the content stored in the brain-function-improving content storage unit for each region according to a predetermined classification method, and submits the selected content to the user interface output unit, the content selection unit that converts the content into a score by a weight for each level, and selects the content so that a sum of the scores in which the weight is reflected for each region is constant; an achievement determination unit executed by the computer or the mobile terminal, the achievement determination unit that determines achievement of the user based on scores acquired through feedback provided by the user for the content selected by the content selection unit; and a content change adjustment unit executed by the computer or the mobile terminal, the content change adjustment unit that controls the content selection unit to newly adjust and select the level when feedback including consecutive correct answers or incorrect answers to at least two items of the selected content is transmitted by the user; an alarm unit that includes at least one of a lighting device, an audio device, and a vibration device; and a sleep mode determination unit executed by the computer or the mobile terminal, the sleep mode determination unit that determines, as a sleep mode, a case in which a specific time is set according to a predetermined period of time and the system is not used by the user after the predetermined period of time; a Global Positioning System (GPS) unit executed by the computer or the mobile terminal, the Global Positioning System (GPS) unit that generates current position information of the user in the case of the sleep mode, and transmits the generated current position of the user; a message generating unit executed by the computer or the mobile terminal, the message generating unit that receives the current position information of the user from the GPS unit, and generates a message with the position information of the user for notifying a caregiver of the user in a case of the sleep mode; a communication unit executed by the computer or the mobile terminal, the communication unit that transmits the message and the current position information to the caregiver of the user; and a mobile device that is separated from the computer or the mobile terminal and is linked to the communication unit; wherein the system registers the identification information in a mobile communication network or an Access Point (AP), and, when the user who carries the mobile device is separated from the system by a predetermined distance or greater, or communication between the mobile device and the system is cut off, the communication unit that receives the identification information of the system from the mobile communication network or the Access Point generates an alarm signal and transmit the generated alarm signal to the alarm device of the system.