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Internal energy is a state function because it merely measure the energy difference that the system holds from one time to another; since internal energy is not affected by how it gets from one state to another, it is a state function.
Internal energy is a state function because it depends only on the current state of the system. For example, It doesn't matter whether you added 50J of heat or the surroundings did 50J of work on the system. Therefore, it is independent of how the system achieved the current state.
It is like the hill analogy. The path you take to get to the top does not matter, as you only look at the change in elevation from your starting point to the end point. Therefore, internal energy is not concerned with the "in between", only the beginning and final states.
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