Matrix exponents and nilpotent algebras

The differentiable at zero function $f$ acting in the matrix algebra $\mathrm M_n(\mathbb C)$ ($n\in\mathbb N$, $n>1$) with the properties $f(X+Y)=f(X)f(Y)$ and $f(0)=I$ is studied. The theorems about the general form of such functions are proved.