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Let $f$ be a function satisfying $f(0)>0, f(1)<0$ , prove that $\exists x_0$ st $f(x_0)=0$ under the assumption that there exists continuous function $g(x)$ such that $f+g$ is non decreasing.

I noticed that if $0\le x\le1$ then $$g(0)<g(0)+f(0)\le g(x)+f(x)\le g(1)+f(1)< g(1)$$ so that $g$ assumes all values $g(x)+f(x)$ for $x\in(0,1)$ using intermediate value property.How shall I proceed further?

Mathronaut
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  • It reminds me of the question from Apostol's Mathematical Analysis which asks: if $f$ is monotonically increasing and $f(0) > 0$ and $f(x) \neq x$ for all $x$ then prove that $f(1) > 1$. – Paramanand Singh Aug 28 '14 at 10:57

1 Answers1

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Hint: Let $x_0=\inf\{x: f(x)\lt0\}$. Consider the possibilities $f(x_0)\lt0$ and $f(x_0)\gt0$ separately, and argue a contradiction for each.

Barry Cipra
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