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Problem Prove that $$\log(1 + \sqrt{1+x^2})$$ is uniformly continuous.

My idea is to consider $|x - y| < \delta$, then show that $$|\log(1 + \sqrt{1+x^2}) - \log(1 + \sqrt{1+y^2})| = \bigg|\log\bigg(\dfrac{1 + \sqrt{1+x^2}}{1 + \sqrt{1+y^2}}\bigg)\bigg| < \epsilon$$ But I couldn't find a choice for $x, y$ that could implies the above expression is true. Completing the square doesn't seem to help at all. Any idea?

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Uniform continuity only makes sense over a given domain. Since this function is continuous, continuity is trivial over any compact set. Are you asked to show this over $\mathbb R$? – Patrick Da Silva Dec 4 '12 at 1:13
@PatrickDaSilva: Yes, I want to show this function is uniformly continuous over $\mathbb{R}$. – Chan Dec 4 '12 at 4:19
up vote 4 down vote accepted

The derivative of $f(x)=\log(1 + \sqrt{1+x^2})$ is $\frac{x}{1 + x^2 + \sqrt{1 + x^2}}$, which is bounded in the whole real line since it is continuous and tends to $0$ as $x\to\pm\infty$. By the Mean Value Theorem, $f$ is Lipschitz and so uniformly continuous.

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Since $1+x^2\geq 1$ and $\sqrt{1+x^2}\geq |x|$ for all $x$, it follows that $|f'(x)|\leq \dfrac{|x|}{1+|x|}< 1$ for all $x$, giving more specifically $|f(x)-f(y)|<|x-y|$ for all $x$ and $y$ (although the best Lipschitz constant is smaller than $1$). – Jonas Meyer Dec 4 '12 at 1:46
The best Lipschitz constant is $0.300283\cdots$. – lhf Dec 4 '12 at 1:49
Or $\frac{\sqrt{2 \left(-1+\sqrt{5}\right)}}{2+\sqrt{2 \left(3+\sqrt{5}\right)}}$? – Jonas Meyer Dec 4 '12 at 1:55
@JonasMeyer, WA says it's $(\sqrt5-2) \sqrt{1/2 (1+\sqrt5)}$. – lhf Dec 4 '12 at 1:59
They're the same, I just didn't get as pretty a form initially using Mathematica. Also $\sqrt{\frac{1}{2} \left(-11+5 \sqrt{5}\right)}$. – Jonas Meyer Dec 4 '12 at 2:05

Note that $t\mapsto \log t$ is uniformly continuous on $[1,\infty)$ (proven below). Note also that $t\mapsto 1+\sqrt{1+t^2}$ is uniformly continuous on $\mathbb R$ (also proven below). As the composition of uniformly continuous functions is uniformly continuous, the result follows.

To see that $\log$ is uniformly continuous on $[1,\infty)$, fix $\varepsilon>0$. Assume that $1\leq x<y$, $y-x<\delta$. Then $y/x<1+\delta/x\leq1+\delta$, and so $$ \log y-\log x=\log \frac y x\leq\log(1+\delta); $$ if we choose $\delta$ small enough so that $\log(1+\delta)<\varepsilon$, we are done.

For the uniform continuity of $g:t\mapsto 1+\sqrt{1+t^2}$, fix $\varepsilon>0$. Choose $x_0$ such that $\sqrt{1+x^2}>3/\varepsilon$ if $|x|\geq x_0$. Since $g$ is continuous on the compact set $[-x_0-1,x_0+1]$, it is uniformly continuous there. So there exists $\delta_1>0$ such that $x,y\in[-x_0,y_0]$ with $|y-x|<\delta_1$ implies $|g(y)-g(x)|<\varepsilon$.

Now let $\delta=\min\{\delta_1,\varepsilon/3,1\}$. Suppose that $|x-y|<\delta$. If both $|x|<x_0$, then $|y|<x_0+1$ and so $|g(y)-g(x)|<\varepsilon$ by the uniform continuity on the compact set. If $|x|\geq|x_0$, then $$ |g(y)-g(x)|=|\sqrt{1+y^2}-\sqrt{1+x^2}|\leq|\sqrt{1+y^2}-|y||+||y|-|x||+||x|-\sqrt{1+x^2}|\\ \leq\frac1{|y|+\sqrt{1+y^2}}+|y-x|+\frac1{|x|+\sqrt{1+x^2}}<\frac1{\sqrt{1+y^2}}+\frac1{\sqrt{1+x^2}}+\frac\varepsilon3\\ <\frac\varepsilon3+\frac\varepsilon3+\frac\varepsilon3=\varepsilon. $$

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