Does this kind of function have special name?

Question

Assume i have a function (or binary operation) $f(a,b)$ such that $f(a,b_1) \ne f(a,b_2)$ if $b_1 \ne b_2$ and $f(a_1,b) \ne f(a_2, b)$ if $a_1 \ne a_2$. For example addition satisfies this property ($a + b_1 \ne a + b_2$), but multiplication does not ($a_1 \cdot 0 = a_2 \cdot 0$). Notice that if $a_1 \ne a_2$ and $b_1 \ne b_2$ at the same time, than it is possible but not necessary that $f(a_1,b_1) = f(a_2,b_2)$. Does this kind of function (or function property) have special name?