In mathematics and computer science, a partial function from the domain X to the codomain Y is a binary relation over X and Y which associates with every element in the set X at most one element in the set Y. If a partial function associates with every element in its domain precisely one element of its codomain, then it is termed a total function, or simply a "function" as traditionally understood in mathematics. Note that with this terminology, not every partial function is a "true" function.
This above diagram does not represent a "well-defined" function because the element 1 in X is not associated with anything.
The natural logarithm function from the real numbers to the reals is only partial, as the logarithm of non-positive reals is not a real number.
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