In operator algebras, there is something called an asymptotic morphism. If $A$ and $B$ are $C^*$-algebras, an asymptotic morphism is a family of maps $T_h : A \to B$ for $0 < h \le 1$, such that for any $a,b \in A$ and $\lambda \in \mathbb{C}$, $$T_h(a + \lambda b) - T_h(a) - \lambda T_h(b) \to 0$$ $$T_h(a^*) - T_h(a)^* \to 0 \$$ $$T_h(ab) - T_h(a)T_h(b) \to 0$$ as $h \to 0$.
These are used to construct maps in K-theory which don't come from $*$-homomorphisms.