weakly locally one-to-one? Is there any standard name for this concept that is weaker than local one-to-one-ness?
In some open neighborhood of $x_0$ there is no point $x\ne x_0$ such that $f(x)=f(x_0)$.
Or, if you like: In some neighborhood of $x_0$, for every point $x$ in that neighborhood, $f(x)=f(x_0)$ only if $x=x_0$.
Might one simply say that "$f$ is weakly locally one-to-one at $x_0$"?
Trivial question, it might seem, if it's only about mathematics.  Maybe it's about psychology of learning mathematics.  I recently came across this error: If $f\;'(a)>0$, the $f$ is strictly increasing in some neighborhood of $a$.  (Much less recently, I made that mistake myself, when, as an undergraduate, I was frustrated by my inability to write what I thought must be a simple $\varepsilon$-$\delta$ proof of that proposition.)  Would people be less likely to make that mistake if somewhere in the recesses of their brain they subconsciously remembered hearing that "If $f\;'\ne 0$ at a point, then $f$ is weakly locally one-to-one at that point."?
 A: A map between topological spaces is called discrete if the preimage of every point is a discrete set, i.e., has no points of accumulation. [In this post all maps are continuous.] This property is weaker than being locally injective. As you observe, every differentiable function $f:(a,b)\to\mathbb R$ with nonvanishing derivative is discrete.
The discreteness property usually appears in tandem with openness (the image of every open set is open). For example, every discrete and open map between planar domains is the composition of a holomorphic function with a homeomorphism; this is Stoilov's theorem. In higher dimensions the equation (discrete and open)=(X)$\circ$(homeomorphism) remains unsolved for X. (One would like the elements of class X to satisfy some sort of Cauchy-Riemann equations.) 
A fundamental result on discrete and open maps between $n$-dimensional topological manifolds is Chernavskii's theorem: the branch set (=the set of points where the map is not locally injective) has topological dimension $\le n-2$; in particular it does not separate the space. 
