This above diagron does not represent a "welldefined" function; because, the element 1, in X, is associated with nothing.
The Turing Machine Partial functions are often used in theoretical computer science: the behavior of a Turing machine for instance can be described by a partial function relating its inputs to its outputs. This is not in general a total function since a Turing machine does not always produce an output for every input: it can run into an infinite loop. Even worse, it can run into an infinite loop for different inputs.
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