I'm reading Howard Georgi online book on The Physics of Waves and found the following argument. Given the functional equation
$ z(t+a) = h(a)z(t) $
he makes the following derivation (I'm citing the book):
If we differentiate both sides of [the aforementioned equation] with respect to $a$, we obtain $ z'(t+a) = h'(a)z(t). $ Setting $a = 0$ gives $ z'(t) = Hz(t) $ where $ H \equiv h'(0). $ This implies $ z(t) \propto e^{Ht}. $ Thus the irreducible solution is an exponential! [...]
How can he differentiate with respect to $a$ then solve the resulting differential equation with respect to $t$? This makes no sense to me.