Techniques to prove a function is uniformly continuous

Solution 1:

First of all, you make a mistake many beginners make: when talking about uniform continuity, it is essential to tell on what set you are studying it. For instance, $x^2$ is uniformly continuous on $[0,r]$ for every $r>0$, but is not so on $[0,\infty)$. Similarly, $\sin \frac 1 x$ is uniformly continuous on $[r,\infty)$ for every $r>0$, but is not so on $(0,\infty)$. The domain of definition is crucial.

Second, there are two main alternative techniques to plainly using the definition:

• the first one is quick: if $f$ is defined on a compact set and is continuous, then it is uniformly continuous

• the second one is subtler: if $f$ is derivable with $f'$ continuous and you manage to find a $M>0$ such that $\sup |f'(x)| \le M$ on your domain of definition, then $f$ will be Lipschitz and, in particular, uniformly continuous.

Solution 2:

This would be a perfectly fine argument, if your $\delta$ were correct. You really are using the definition of uniform continuity, which is precisely that one can pick for any $\epsilon$ a $\delta$ independent of $x_0$. Unfortunately, your $\delta$ is wrong for $\sqrt x$. If $x=10^{-2},y=10^{-4},$ then certainly $|x-y|<10^{-2}$ and $|x-y|<1$, but $|\sqrt(x)-\sqrt(y)|>10^{-2}$. Your numbers are fine on $[1,\infty)$, but that's by far the easier part!

As for $x^2$, all you've done is show that there is some choice of $\delta$ which depends on $x_0$. That's a far cry from showing that every choice of $\delta$ depends on $x_0$, so you have not shown that $x^2$ fails to be uniformly continuous.

Solution 3:

As you have an answer for the first case, I only give you a sketch for $x^2$.

I leave you to check: $f$ is uniformly continuous iff for all $x_n-y_n\to 0\implies f(x_n)-f(y_n)\to 0$. For the case $x^2$ is as simple as pick two sequences whose difference converges to $0$ but the difference of its images does not.

Consider $x_n=n+1/n$ and $y_n=n$. So $x_n-y_n\to 0$ but $x_n^2-y_n^2=2+1/n^2\not\to0$. Thus $x^2$ is not uniformly continuous.