Studying Set theory, specifically functions. My textbook says as follows:
A function $f:A\rightarrow B$ is a relation from $A$ to $B$ (i.e, a subset of $A\times B$) such that each $a$ in $A$ belongs to a unique ordered pair, $(a,b)$ in $f$.
This seems to add an additional 'rule' to the definition of a function from what I am familiar with in calculus. The requirement that every $a$ must map to some b is confusing to me. What about functions with vertical asymptotes? Don't those functions map from $\mathbb R\rightarrow\mathbb R$, but there is some a value where the function is not defined? Or the same could be said about functions with "holes".
Any help would be appreciated. Thanks!
from Hot Weekly Questions - Mathematics Stack Exchange
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