Suppose $f$ is uniformly continuous on $X_1,X_2,...,X_n$.
Let $X=\cup_{i=1}^n X_i.$
a.) Show that $f$ need not be continuous on $X$.
b.) Show that $f$ need not be uniformly continuous on $X.$
This task seems daunting, but let me know what you think. I'm struggling.
For a.), I could imagine constructing a union of disjoint open intervals and make the function undefined at each endpoint.
Here's what I mean:
Let $X=(0,a)\cup(a,b)\cup(b,c)\cup...\cup(y,z)$.
Let $f:X\to\mathbb{R}$ be given by $f(x)=\frac{1}{(x-a)(x-b)(x-c)...(x-y)(x-z)}$
Now, for any $x \neq a,b,c,...,y,z, f(x)$ is continuous
and for any $x=a,b,c,...,y,z, f(x)$ is undefined
But do note that for any $X_i, f(x)$ is continuous on the $X_i$ interval
For b.), I'm not sure where to start. In many cases, uniform continuity deals with vertical asymptotes so it's hard to imagine a union of sets that don't have vertical asymptotes, but one of their individual sets does.