No. For example, consider the discontinuous bijection that increases linearly from [0,0] to [1,1], decreases linearly from (1,2) to (2,1), increases linearly from [2,2] to [3,3], decreases linearly from (3,4) to (4,3), etc.
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By definition, a permutation is a bijection from a set to itself. Since a permutation is bijective, it is one-to-one.
It can be quite hard. First determine the domain. Then, for every input value from the domain, calculate the output value. The set of all these output values is the range. For simple functions you will not need to find every output value. For monotonic continuous functions the end points of the domain will determine the endpoints of the range. [Monotonic means never decreasing or never increasing]. For non-monotonic functions, for example a quadratic or polynomial of higher order, you may need to find the turning points.
Not every relation is a function. But every function is a relation. Function is just a part of relation.
Every function differs from every other function. Otherwise they would not be different functions!
That is part of the definition of a function.