Prove that the function $f: (0,1) \to \Bbb{R}$ defined by $f(x) = \frac{1}{x}$ is not uniformly continuous.

On the contrary, we assume there is a $\delta$ and we derive a contradiction.

Let $a_n = \frac{1}{n}$ and $b_n = \frac{1}{n+1}$. We observe that $|\frac{1}{n} - \frac{1}{n+1}|$ can be made arbitrarily small as the sequence $\{\frac{1}{n}\}$ converges to 0. Hence $|\frac{1}{n} - \frac{1}{n+1}|$ can be made less that $\delta$ for large $n$. Now, for such $n$, $|f(\frac{1}{n})-f(\frac{1}{n+1})|$ should be less than $\epsilon = \frac{1}{2}$. But $$|f(\frac{1}{n})-f(\frac{1}{n+1})| = |n -(n+1)| = 1$$ contradiction!.

The general strategy is, to prove a function is not uniformly continuous, find two sequences $a_n$ and $b_n$ such that $|a_n - b_n|$ is arbitrarily small for large $n$ but $|f(a_n) - f(b_n)|$ is a fixed constant for all $n$. Why it is enough to prove such sequences is explained above using $\epsilon$ and $\delta$ argument.

**Proof**: A function $f : (0,1) \to \Bbb{R}$ is said to be uniformly continuous if given $\epsilon > 0$ there exists a $\delta>0$ such that $|x-y| < \delta$ implies that $|f(x) - f(y)| < \epsilon$**for all**$x,y \in (0,1)$. We will show that, for $\epsilon = \frac{1}{2}$ finding such a $\delta$ (which works**for all**$x,y \in (0,1)$) is not possible.On the contrary, we assume there is a $\delta$ and we derive a contradiction.

Let $a_n = \frac{1}{n}$ and $b_n = \frac{1}{n+1}$. We observe that $|\frac{1}{n} - \frac{1}{n+1}|$ can be made arbitrarily small as the sequence $\{\frac{1}{n}\}$ converges to 0. Hence $|\frac{1}{n} - \frac{1}{n+1}|$ can be made less that $\delta$ for large $n$. Now, for such $n$, $|f(\frac{1}{n})-f(\frac{1}{n+1})|$ should be less than $\epsilon = \frac{1}{2}$. But $$|f(\frac{1}{n})-f(\frac{1}{n+1})| = |n -(n+1)| = 1$$ contradiction!.

The general strategy is, to prove a function is not uniformly continuous, find two sequences $a_n$ and $b_n$ such that $|a_n - b_n|$ is arbitrarily small for large $n$ but $|f(a_n) - f(b_n)|$ is a fixed constant for all $n$. Why it is enough to prove such sequences is explained above using $\epsilon$ and $\delta$ argument.

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