A strongly continuous family \(\{S(t);t\geq 0\}\) of bounded linear operators on a Banach space \(X\) is called a \(C\)-semigroup if it satisfies: \(S(0)=C \text{ and } S(s)S(t)=S(s+t)C \text{ for } s, t\geq 0.\) A strongly continuous family \(\{C(t);t\geq 0\}\) of bounded linear operators on \(X\) is called a \(C\)-cosine function if it satisfies: \(C(0)=C \text{ and } 2C(t)C(s) = [C(t+s) + C(|t-s |)]C \text{ for } s, t \geq 0.\) A linear operator \(T\) is said to be Hermitian or positive if it has numerical range \(V(T):=\{f(T); f\in B(X)^*, \|f\|=f(I)=1\}\) contained in the real line \(\mathbb{R}\) or in \([0,\infty)\), respectively. The following statements are proved in this paper: (1) \(C\)-semigroups of Hermitian operators are infinitely differentiable in operator norm on \((0,\infty)\), but not necessarily at \(0\); (2) \(C\)-cosine functions of Hermitian operators are norm continuous at either none or all of points in \([0,\infty)\); (3) \(C\)-semigroups of positive operators which dominate \(C\) are infinitely differentiable in operator norm on \([0,\infty)\); (4) \(C\)-cosine functions of positive operators are infinitely differentiable in operator norm on \([0,\infty)\).