If g is a real-analytic function defined near x<sub>0</sub> with g(x<sub>0</sub>) = x<sub>0</sub> and 0 < λ &ne; 1 where λ := g'(x<sub>0</sub>), then Koenigs proved that there exists a real-analytic homeomorphism h defined near x<sub>0</sub> such that hgh<sup>-1</sup>(x) = L(x) where 

L(x) := x<sub>0</sub> + λ(x-x<sub>0</sub>)

and h is unique up to a constant factor*. 

This allows g to be embedded in a local flow: Φ(x,t) := h<sup>-1</sup> L<sup>t</sup> h(x), where 

L<sup>t</sup>(x) := x<sub>0</sub> + λ<sup>t</sup> (x-x<sub>0</sub>),

such that Φ(x,1) = g(x) where defined.

Then Φ satisfies the differential equation dx(t)/dt = V(x(t)) where the velocity V (denoted by f in the Question) is given by

V(x) := ∂Φ(x,t)/∂t |<sub>t=0</sub>.

Note, however, that the flow Φ(x,t) into which the original function g embeds as the time-1 map need not be unique when there exist more than one fixed point of g.  As an example, for x > 0 consider the function 

g<sub>c</sub>(x) := c<sup>x</sup>. 

Then for 1 < c < e<sup>1/e</sup> the function g<sub>c</sub> has two distinct fixed points each satisfying the hypotheses of the Koenigs theorem, and these give two distinct flows into which g<sub>c</sub> embeds as the time-1 map.  

Concretely, set c = &radic;2, so that g<sub>c</sub>(x) = x for both x = 2 and x = 4, with derivatives 

g<sub>c</sub>'(2) = ln(2) and 

g<sub>c</sub>'(4) = ln(4).  

Calculating the respective real-analytic flows Φ<sub>2</sub>(x,t) and Φ<sub>4</sub>(x,t), both are defined for (x,t) = (3, 1/2). 

But Φ<sub>2</sub>(3, 1/2) and Φ<sub>4</sub>(3, 1/2) first differ in the 25th decimal place. Hence they are solutions of distinct differential equations. I.e., they have different velocity functions.
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&#42; This is actually true in greater generality; see J. Milnor's book *Dynamics in One Complex Variable*, 3rd ed., Princeton University Press, 2006.