Prove that the function is uniformly continuous in $ \mathbb{R}$.

Question

$f(x)=(\sin x+\cos x)^{100}$ in $ \mathbb{R}$.

(My attempt)

I first thought that it is not uniformly continuous.

Using $\sin x+\cos x=\sqrt{2}\sin\left(x+\frac{\pi }{4}\right)$,

then $f(x)=2^{50}\left(\sin\left(\frac{\pi}{4}\right)\right)^{100}$

Let $\epsilon > 0 $ be given.

For every $\delta > 0$, let $ x=2n\pi -\frac{\pi }{4}+\frac{\delta }{2} $

and $ y=2n\pi -\frac{\pi }{4}-\frac{\delta }{2}$ for any $n\in \mathbb{N}$.

That is $\left \lvert x-y \right\rvert< \frac{\delta }{2}< \delta $. Then $\left \lvert f(x)-f(y) \right \rvert=2^{50}\left \lvert \sin\left(x+\frac{\pi }{4}\right)-\cos\left(x+\frac{\pi }{4}\right) \right \rvert\left \lvert \left(\sin\left(x+\frac{\pi }{4}\right)\right)^{99}+ \dots +\left(\cos(x+\frac{\pi }{4}) \right)^{99} \right \rvert=2^{51}\left \lvert \sin\left(\frac{\delta }{4}\right) \right \rvert\left \lvert 100\left(\sin\left(\frac{\delta }{4}\right)\right)^{99} \right |=200\cdot 2^{50}\left \lvert \left(\sin\left(\frac{\delta }{4}\right)\right)^{100}\right \rvert> 1=\epsilon $.

So, it is contradiction. Hence, $f$ is not uniformly continous in $ \mathbb{R}$.

But, the answer to this problem is that $f(x)$ is uniformly continuous $ \mathbb{R}$.

I don't know why this is uniformly continuous.

Answer 1

The function $f$ is continuous Therefore, $f|_{[0,2\pi]}$ is uniformly continuous and, since $f$ is also periodic, with period equal to $2\pi$, it is not hard to deduce that $f$ is indeed uniformly continuous.

Answer 2

Use the following facts:

1. $a^n-b^n=(a-b)\left(\sum_{j=0}^{n-1}a^jb^{n-j-1}\right)$. Thus $$\begin{aligned}&|\sin^{100}(x+\pi/4)-\sin^{100}(y+\pi/4)|\\ =&|\sin(x+\pi/4)-\sin(y+\pi/4)|\left|\sum_{j=0}^{99}\sin^{j}(x+\pi/4)\sin^{99-j}(y+\pi/4)\right|\\ \leq &100|\sin(x+\pi/4)-\sin(y+\pi/4)|.\end{aligned}$$
   2. Next, use the identity $$\sin a-\sin b=2\cos\frac{a+b}{2}\sin\frac{a-b}{2}.$$
   3. Finally , use the fact that $$|\sin x|\leq|x|$$ for all real number $x$.

Answer 3

$|f'(x)|\leq 200(2^{99})$ so $f$ is uniformly continuous by MVT.