Uniform Continuity for 1/x

**AKTilted** · November 4th 2010, 12:29 AM

I'm trying to prove the following statement:

Prove that the function f(x)=1/x is continuous on (0,1] but it is not uniformly continuous on this interval.

Thanks a lot.

**Tinyboss** · November 4th 2010, 05:20 AM

For the "not uniformly continuous" part: do you know how to formally negate a logical proposition, i.e. exchange "for-all's" and "there-exists's" and reverse (in)equalities? Try that with the definition of uniform continuity.

**Drexel28** · November 4th 2010, 06:04 AM

Originally Posted by AKTilted

I'm trying to prove the following statement:

Prove that the function f(x)=1/x is continuous on (0,1] but it is not uniformly continuous on this interval.

Thanks a lot.

I actually discuss this in my post here. In essence, (we'll speak less generally now) if $f<img src=$ (0,1]\to\mathbb{R}" alt="f

(0,1]\to\mathbb{R}" /> were uniformly continuous, then we could extend it to some uniformly continuous map $\tilde{f}:[0,1]\to\mathbb{R}$ . But, this is just nonsense since we'd have to have that $f(0)=\lim f\left(\frac{1}{n}\right)=\lim n$

**Also sprach Zarathustra** · November 4th 2010, 06:08 AM

Originally Posted by AKTilted

I'm trying to prove the following statement:

Prove that the function f(x)=1/x is continuous on (0,1] but it is not uniformly continuous on this interval.

Thanks a lot.

The negation of uniform continuity:

For some $\epsilon > 0$ and for each $\delta > 0$
there exist $x_0, x_1$ such that $|x_0 - x_1| < \delta$ but $|f(x_0) - f(x_1)| \geq \epsilon$ .

Assume that there is exist $\delta <\frac{1}{n}$ $n\in \left \{\mathbb{N} / 0}{ \right \}$ .

Now let $x_0,x_1 \in\mathbb{R}^+ x_0=x_1+\frac{\delta }{2} and x_1=\frac{\delta }{2}$

$\left | f(x_0)-f(x_1) \right |=\left | \frac{1}{x_0}-\frac{1}{x_1} \right |=\left |\frac{ x_0-x_1}{x_0x_1} \right |<br />$

$=\frac{\frac{2}{n}}{x_0(x_0+\frac{\delta }{2})}\frac{\delta }{2}=\frac{\frac{2}{n}}{\delta^2}\frac{\delta }{2}=\frac{\frac{1}{n}}{\delta }>1$

**Also sprach Zarathustra** · November 4th 2010, 06:15 AM

Another much better solution:

Choose $x_n=\frac{1}{n}$ and $y_n=\frac{1}{n+1}$ for any $n\in\mathbb{N}$ .

$\left | x_n-y_n \right |<\frac{1}{n}$ , but $\left | f(x_n)-f(y_n) \right |=1<br />$ .

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