  Let $A$ and $B$ be C$^*$-algebras. A linear map $T:A\to B$ is said to be a
$^*$-homomorphism at an element $z\in A$ if $a b^*=z$ in $A$ implies $T (a b^*)
=T (a) T (b)^* =T(z)$, and $ c^* d=z$ in $A$ gives $T (c^* d) =T (c)^* T (d)
=T(z).$ Assuming that $A$ is unital, we prove that every linear map $T: A\to B$
which is a $^*$-homomorphism at the unit of $A$ is a Jordan $^*$-homomorphism.
If $A$ is simple and infinite, then we establish that a linear map $T: A\to B$
is a $^*$-homomorphism if and only if $T$ is a $^*$-homomorphism at the unit of
$A$. For a general unital C$^*$-algebra $A$ and a linear map $T:A\to B$, we
prove that $T$ is a $^*$-homomorphism if, and only if, $T$ is a
$^*$-homomorphism at $0$ and at $1$. Actually if $p$ is a non-zero projection
in $A$, and $T$ is a $^*$-homomorphism at $p$ and at $1-p$, then we prove that
$T$ is a Jordan $^*$-homomorphism. We also study bounded linear maps that are
$^*$-homomorphisms at a unitary element in $A$.
