Help proving a theorem on uniform continuity in an open interval

Question

I need help proving that given f:(a,b)→R that is uniformly continuous, it is possible to extend f to f:[a,b]→R that is continuous on the closed interval.

thanks in advance!

Answer 1

Hints using the comment by David:

Show that the limits

$$\lim_{x\to a^+}f(x)\;\;,\;\;\;\;\lim_{x\to b^-}f(x)$$

exist finitely, and then extend the definition of $\;f\;$ on those end points accordingly.

For example: if $\;\epsilon>0\;$ , we know that there exists $\;\delta>0\;$ s.t.

$$\left(x,y\in (a,b)\;\;\wedge\;\;|x-y|<\delta\right)\;\implies\; |f(x)-f(y)|<\epsilon$$

The above is true also in any right neighborhood of $\;a\;$ (contained in $\;(a,b)\;$, of course), so deduce that for any sequence

$$\;\left(\{x_n\}_{n\in\Bbb N}\subset (a,b)\;\;s.t.\;\;x_n\to a\right)\implies \{f(x_n)\}_{n\in\Bbb N}$$

converges finitely since it is a Cauchy sequence, say to $\;L\;$ , and then just put $\;f(a):=L\;$ .