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Let $G\neq \emptyset$ be a set, and $\lambda\in G$. Define $D(t,\lambda):[a,b]\to \mathbb{R}$, continuous. My queston is: is there a function $x:[a,b]\to \mathbb{R}$, such that:

  1. $x$ is continuous and $\|x\|=1$
  2. $x(t)D(t,\lambda)\geq 0$ for all $t\in[a,b]$?

Any help will be appreciate! This is a conclusion that I need to prove another problem, but I don't know if is possible!!
Thanks

Valent
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  • cant you just take $x(t):=D(t,\lambda)/||D(\cdot,\lambda)||$? – sranthrop Jun 21 '15 at 18:17
  • Thanks, this works for $|D(t,\lambda)|\neq 0$ can we fix this for the points in where $D(t,\lambda)=0$? – Valent Jun 21 '15 at 18:40
  • If $||D(\cdot,\lambda)||=0$, then $D(t,\lambda)=0$ for all $t$. Note that $||\cdot||$ is a norm (I think it is the max-norm?) – sranthrop Jun 21 '15 at 18:45
  • Can you please help me with this problem? Is the related problem! http://math.stackexchange.com/questions/854333/a-problem-in-the-space-ca-b – Valent Jun 21 '15 at 19:06

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