total function

let $\text{[math]}$ be non-empty sets and let $\text{[math]}$ be a relation from $\text{[math]}$ to $\text{[math]}$
we say $\text{[math]}$ is a total function from $\text{[math]}$ to $\text{[math]}$ if for every $\text{[math]}$ there exists a single $\text{[math]}$ such that $\text{[math]}$ , we write $\text{[math]}$ and $\text{[math]}$ instead of $\text{[math]}$

note that $\text{[math]}$ is a necessary condition for $\text{[math]}$ to be total

consider $\text{[math]}$ defined as $\text{[math]}$
this function isnt defined at $\text{[math]}$ therefore there is an $\text{[math]}$ that doesnt have a corresponding $\text{[math]}$ therefore this function isnt a total function but rather a partial function
it is however a total function if we were to take the domain minus the discontinuity, i.e. $\text{[math]}$