limits and uniformly continuous

**inthequestofproofs** · March 24th 2010, 07:36 PM

I need to show that if f is uniformly continuous on (a,b], then the right hand limit of f(x) exists.

This is what i have so far:
If f is uniformly continuous on (a,b], given epsilon >0, there is a delta > 0 s.t. |x-y| < delta -> |f(x)-f(y)| < epsilon for all x,y in (a,b]
Let x be in (a, a + delta).
|f(x) - f(y)|
<= |f(x) - L|+|L-f(y)|
< |f(x) - L|
< epsilon.

Thus, the right hand limit of f exist

Does this make sense?

**Laurent** · March 25th 2010, 04:47 AM

Originally Posted by inthequestofproofs

I need to show that if f is uniformly continuous on (a,b], then the right hand limit of f(x) exists.

This is what i have so far:
If f is uniformly continuous on (a,b], given epsilon >0, there is a delta > 0 s.t. |x-y| < delta -> |f(x)-f(y)| < epsilon for all x,y in (a,b]
Let x be in (a, a + delta).
|f(x) - f(y)|
<= |f(x) - L|+|L-f(y)|
< |f(x) - L|
< epsilon.

Thus, the right hand limit of f exist

Does this make sense?

Not really. First, we don't know what your $L$ is; second, $|f(x)-L|+|L-f(y)|<|f(x)-L|$ is more than dubious.

Since you have to prove the existence of a limit and your function is not monotone, you should use Cauchy condition. This should be pretty straightforward (no computation at all).

**inthequestofproofs** · March 25th 2010, 07:32 AM

sorry for so many questions, but the cauchy condition is for sequences. I don't know how to apply it for a function. Could you explain?

**Laurent** · March 25th 2010, 11:13 AM

Originally Posted by inthequestofproofs

sorry for so many questions, but the cauchy condition is for sequences. I don't know how to apply it for a function. Could you explain?

You have the same criterion as for sequences.

Fact (Cauchy property for functions) : $f(t)$ has a limit when $t\to a^+$ if, and only if, for every $\epsilon>0$ , there is $t_0>a$ such that, for any $t,t'\in(a,t_0)$ , $|f(t)-f(t')|<\epsilon$ .

Proof:
a) Assume $f(t)\to_{t\to a^+}\ell$ . Let $\epsilon>0$ . There is $t_0>a$ such that, if $a<t<t_0$ , $|f(t)-\ell|<\epsilon/2$ . Then, for all $a<t,t'<t_0$ , $|f(t)-f(t')|\leq |f(t)-\ell|+|\ell-f(t')|\leq \epsilon$ .

b) Assume Cauchy condition : for every $\epsilon>0$ , there is $t_0>a$ such that, for any $a<t,t'<t_0$ , $|f(t)-f(t')|<\epsilon$ . Choose a particular sequence $(u_n)_{n\in\mathbb{N}}$ such that $u_n\to_n a^+$ . Then the assumption shows that $(f(u_n))_{n\in\mathbb{N}}$ is a Cauchy sequence (I let you check that). Therefore, it has a limit $\ell\in\mathbb{R}$ : $f(u_n)\to_n \ell$ .
Let us show that $f(t)\to_{t\to a^+} \ell$ . Let $\epsilon>0$ . Let $t_0$ be given by the assumption. There is $n_0$ such that $n\geq n_0$ implies $a<u_n<t_0$ (because $u_n\to_n a^+$ ) and $|f(u_n)-\ell|<\epsilon$ (because $f(u_n)\to_n\ell$ ). For every $a<t<t_0$ , we have $|f(t)-\ell|\leq |f(t)-f(u_n)|+|f(u_n)-\ell|\leq 2\epsilon$ (first term because of Cauchy condition since $a<u_n,t<t_0$ ). This concludes.

--
In your problem, the uniform continuity immediately gives the Cauchy condition.

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