# Math Help - Existence of a root

1. ## Existence of a root

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

2. Originally Posted by NonCommAlg
We have two functions $f,g : [a,b] \longrightarrow \mathbb{R}$ such that $g$ is continuous, $f+g$ is decreasing, $f(a) < 0$ and $f(b) > 0.$ Prove that there exists $a < c < b$ such that $f(c) = 0.$
Binary search will converge to a point $c$ such that for all $0<\varepsilon<\varepsilon_0$ (that is for all $\varepsilon$ small enough):

$
f(c-\varepsilon)<0$
and $f(c+\varepsilon)\ge 0
$

Thus either $f(c)=0$ or there is an increasing jump discontinuity at $c$. But an increasing jump at c is imposible given that g(x) is continuous and f(x)+g(x) and decreasing at c.

CB