Unique continuous complex log of a function nowhere zero Consider a function $\phi : \mathbb{R}^d \rightarrow \mathbb{C}$ which is continuous, satisfies $\phi(0)=1$ and is nowhere zero. 
I am reading a book where the following claim is made:
Fix $z \in \mathbb{R}^d$. Let $\log$ denote the complex logarithmic function (multi-valued).  Then there is a unique branch $h_z(t), 0 \leq t \leq 1$ of $\log \phi(tz), 0 \leq t \leq 1$ with the property that $h_z(0) = 0$ and such that $h_z(t)$ is continuous in $t$.
I don't understand why a continuous branch exists (why should the path $t \mapsto \phi(tz), t \in [0,1]$ not wind around the origin?). If this path doesn't wind around the origin, won't there be infinitely many possible choices of rays leaving the origin to use as branch cuts?
Many thanks for your help.  
 A: Forget about branch cuts and work with covering spaces. Consider the map $f:\mathbb{C}\to\mathbb{C}^*$ given by $f(z)=\exp(z)$. This is the universal covering space for $\mathbb{C}^*$.
Now consider the continuous path $\gamma:[0,1]\to\mathbb{C}^*$ given by $\gamma(t)=\phi(tz)$ for some fixed $z\in\mathbb{R}^d$.
By the lifting property of the universal covering, given $p\in\mathbb{C}^*$ such that $\exp(p)=\gamma(0)=1$, there exists a unique continuous lift $\widetilde{\gamma}_p:[0,1]\to\mathbb{C}$ such that $\exp(\widetilde{\gamma}_p(t))=\gamma(t)$ for all $t\in[0,1]$ and $\widetilde{\gamma}_p(0)=p$.
If you take $p=0\in\mathbb{C}$, you get the thesis.
N.B. The winding number is to be considered if you have a loop ($\gamma(0)=\gamma(1)$) and you want to lift it as a loop. Here the only request is that the lifting should be continuous and starting from $0$, so you just look at it as an arc.

Alternative approach
Remark Consider the $1$-form $\omega=\dfrac{dz}{z}$. Then, $\log(w)$, for $w\in \mathbb{C}^*$, can be written as
$$\int_\gamma\omega$$
for a suitable $\gamma:[0,1]\to\mathbb{C}^*$ such that $\gamma(0)=1$ and $\gamma(1)=w$. Depending upon the number of times that $\gamma$ winds up around $0$, you will get the different values of $\log(w)$, changing for an integer multiple of $2\pi i$ (the precise multiple is given by the winding number).
Now, set $g(t)dt=\gamma^*\omega$, i.e. 
$$g(t)=\dfrac{\langle z, \nabla \phi(tz)\rangle}{\phi(tz)}\;.$$
We obtain a function $g:[0,1]\to\mathbb{C}$ (as $\phi(tz)\neq 0$ for every $t$ and $z$). This function is continuous on $[0,1]$, hence integrable.
Now just take a primitive $G$ of $g$ on $[0,1]$ such that $G(0)=0$. $G:[0,1]\to\mathbb{C}$, by the Remark above, is the "logarithm" of $\gamma$.
