Is a function a digraph?

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If I recall correctly, a function is a set of $2$-tuples with some properties, which is itself a digraph if we let the set of nodes be the union of the domain and image of the function. Is this true?

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Yes, it is true.

In fact any binary relation could be considered as a digraph. For example the relation $$ R = \text { { (1,2),(1,3),(3,5), (1,5)}}$$ which is not a function could also be considered as a digraph.