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Let $f\colon \mathbb{R}_{+} \to \mathbb{R}$ uniformly continuous. Prove that exists $K>0$ such that for each $x\in \mathbb{R}_{+},$ $$\sup_{w>0}\{ |f(x+w) -f(w)|\}\le K \,\, ( x + 1).$$

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up vote 1 down vote accepted

I will give you hint instead of a complete answer since it's a homework question.

  1. Let $F\colon\mathbb R_+\to\mathbb R$ defined by $\displaystyle F(x)=\sup_{t>0}|f(x+t)-f(x)|$. Show that $F$ is uniformly continuous on $\mathbb R$.
  2. Let $h\colon \mathbb R_+\to\mathbb R_+$ an uniformly continuous function on $\mathbb R_+$. Prove that we can find a constant $K>0$ such that $h(x)\leq K(x+1)$ for all $x\geq 0$.
  3. Conclude.
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