Constructing smooth paths between points in codomain

I have a differentiable non-bijective mapping f from $x \in R^T$ to $y = f(x) \in R^T$.

Say I have some positions $x_s^*$ and $x_e^*$ with corresponding mappings $y_s*$ and $y_e^*$.

Questions:

1. I assume there exists a smooth path (i.e. $[y_s*, y_2, y_3, ..., y_e^*]$) between $y_s*$ and $y_e*$?
2. How would I go about constructing a such a possible path where for every point $y_i \in R^T$ there exists a point $x_i \in R^T$?
3. If the above is not straight-forward, would it be under certain constraints?