HW6: Challenge 1

Assume that $g$ is defined on an open interval $(a,c)$ and it is known to be uniformly continuous on $(a,b]$ and $[b,c)$ where $a<b<c$ . Prove that $g$ is uniformly continuous on $(a,c)$ .

Assume $g$ is defined on an open interval $(a,c)$ . Suppose $g$ is uniformly continuous on $(a,b]$ and $[b,c)$ with $a<b<c$ .

Suppose $x\in (a,b]$ . Since g is uniformly continuous on $(a,b]$ , we know that for any $\epsilon>0$ , $\exists\delta_1>0$ s.t for $x,b\in(a,b]$ , $\mid x-b\mid<\delta_1$ implies $\mid g(x)-g(b)\mid<\frac{\epsilon}{2}$ .

Similarly suppose $y\in [b,c)$ . Since $g$ is uniformly continuous on $[b,c)$ , we know that for any $\epsilon>0$ , $\exists\delta_2>0$ s.t. for $y,b\in [b,c)$ , $\mid y-b\mid<\delta_2$ implies $\mid g(y)-g(b)\mid<\frac{\epsilon}{2}$ .

Choose $\delta=\min\{\delta_1,\delta_2\}$ so that $\mid x-y\mid<\delta$ implies $\mid x-b\mid<\delta$ and $\mid y-b\mid<\delta$ . Thus, applying the triangle inequality, for all $x,y\in (a,c)$ , if $\mid x-y\mid<\delta$ , it follows that