physical chemistry - What is the exact definition of state functions? How to determine state functions?

How do you know if something is a state function? I know Internal Energy (U) is a state function since it is the sum of energies, which is constant at a given time.

However I don't know for sure if $\Delta$U is a state function. I know that $\Delta$U$= U_{final}-U_{initial}$, and so any path from final to initial will yield the same $\Delta$U. So by this definition any $\Delta$ function should be a state function.

As a result I would expect Heat ($q$) to also be a state function since $q=\Delta$TE, but this is not the case.

Where am i going wrong in my understanding? I understand the "altitude" analogy, however i need a more technical definition with chemistry terms. ]