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Temperature is a state function as it is one of the values used to define the state of an object. Furthermore, temperature is dependent on the final and initial values, not on the path taken to establish the values.
Herein, why is temperature not a state function?
Temperature is a state function. Heat and work are not state functions. Work can't be a state function because it is proportional to the distance an object is moved, which depends on the path used to go from the initial to the final state. If work isn't a state function, then heat can't be a state function either.
Secondly, why heat is a path function?
Heat, the thermodynamic quantity and the change associated with it is a path function. i.e. it depends on how the system changes from 1 state to another. Since the way in which heat is transferred depends not on the path taken, isothermal or isobaric etc., the very concept of the path or point function is absurd.
Pressure and volume change while the temperature remains constant. Since no work or heat are exchanged with the surrounding, the internal energy will not change during this process. Thus, the internal energy of an ideal gas is only a function of its temperature.