Given $k$, $f(x) = e^{x_1}+e^{x_1+x_2}+\cdots+e^{x_1+x_2+\cdots+x_k},$ $x=(x_1,x_2,\ldots,x_k),$ where $x_i \in \{0,1\}$.

We want to compute: $\inf_{x \neq y}|f(x)-f(y)|$ or a lower bound of $\inf_{x \neq y} |f(x)-f(y)|$.

Now I just know that $f$ is an injection when $x$ is rational. Anyone know if there is an analytic solution of this?