A function $f:A \subseteq \mathbb{R} \rightarrow \mathbb{R}$ is uniformly continuous if $\forall \epsilon>0$ $\exists \delta>0$ such that $\forall x \in A and \forall y \in A$ with $|x-y|\leq \delta$ we have $|f(x)-f(y)|\leq \epsilon$
Is the negation:
There exists and $\epsilon>0$ such that $\forall \delta>0$ there exists $x\in A$ or there exists $y\in A$ such that $|x-y|\leq \delta$ and $|f(x)-f(y)|\geq \epsilon$?
Show that $f:(0,1)\rightarrow \mathbb{R}$ given by $f(x)=\frac{1}{x}$ is not uniformly continuous.
proof: Let $\epsilon=1$ let $\delta>0$ arbitrary. Set $x\in (0,1)$ to be such that $x<\frac{\delta}{1+\delta}$ and $y= x+\delta$ . Then $|\frac{1}{x}-\frac{1}{x+\delta}|$ $=$ $|\frac{\delta}{x(x+\delta)}|$ $\geq$ $\frac{\delta}{x(1+\delta)}>1$
Is the proof correct? The only question I have is why is $y=x+\delta$ guaranteed to be in $(0,1)$?
Best Answer
You basically have it. You just need to be a little more careful. You are claiming that for $\epsilon=1$, and for every $\delta>0$, there are $x,y\in (0,1)$ such that$|x-y|<\delta$ and $|f(x)-f(y)|>1.$ You may assume without loss of generality that $\delta<1/2$ because it the claim is true for all such $\delta,$ it will be true for any value of $\delta$ larger than $1/2.$
(Remember, all you need to do is find two numbers in $(0,1)$ whose difference is less than $\delta$ in absolute value. The ones that work for $\delta<1/2$ will also work for $\textit{any}\ \delta\ge 1/2.$ Example: suppose we have $\delta=15$ and you can find $x,y$ such that $|x-y|<1/2$ and $|f(x)-f(y)|>1.$ Then, the $x,y$ work for $\textit{both}$ values of $\delta$ simultaneously because if $|x-y|<1/2$ it is also $<15$).
Now, $|f(x)-f(y)|=\left|\frac{x-y}{xy}\right|$ and we want to choose $x$ and $y$ so that $|x-y|<\delta$ but $\left|\frac{x-y}{xy}\right|>1$, so take $x=\delta$ and $y=2\delta.$ Then, $x$ and $y$ are actually in $(0,1)$ and $\left|\frac{x-y}{xy}\right|=\frac{2}{\delta}>1$, and you are done.
It may be easier to do it with sequences: with $\epsilon=1/2,$ take $\delta_n=1/n$ and find sequences $(x_n)$ and $(y_n)$ such that $|x_n-y_n|\to 0$ but $|f(x_n)-f(y_n)|>1/2.$ Choose $x_n=1/n$ and $y_n=1/n+1$ and check that this assignment works.