We have two functions $\displaystyle f,g : [a,b] \longrightarrow \mathbb{R}$ such that $\displaystyle g$ is continuous, $\displaystyle f+g$ is decreasing, $\displaystyle f(a) < 0$ and $\displaystyle f(b) > 0.$ Prove that there exists $\displaystyle a < c < b$ such that $\displaystyle f(c) = 0.$