Given an arbitrary differentiable scalar function $a : \mathbb{R}^n \to \mathbb{R}$ and an arbitrary differentiable vector field $b: \mathbb{R}^n \to \mathbb{R}^n$, is it always possible to find a scalar differentiable function $c : \mathbb{R}^n \to \mathbb{R}$ such that: \begin{align} \frac{a(x)}{c(x)}+\frac{\nabla c(x)^T b(x)}{c(x)} \leq \sigma \end{align} for some positive real number $\sigma$?
My intuition is that this should be possible by choosing $c(x)$ such that $\frac{\nabla c(x)^T b(x)}{c(x)}$ is arbitrarily small and choosing the sign of $c(x)$ so that $\frac{a(x)}{c(x)}$ is negative. However, I am unable to prove this formally. Any ideas?