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## Homework Statement

(i) Let U and V be open subsets of C with a function f defined on [tex] U \cup V[/tex] suppose that both restrictions, [tex] f_u \mathrm{and} f_v [/tex] are continuous. Show that f is continuous.

(ii) Illustrate by a specific example that this may not hold if one of the sets U, V is not open.

## Homework Equations

## The Attempt at a Solution

Now I'm pretty sure I can prove the first part. As, since f is continuous on U and V then there exists a [tex] \delta_1 [/tex] and on U there will be a [tex] \delta_2 [/tex] and then taking the minimum of these will put [tex] |f(z) - f(w)| < \epsilon [/tex] for all epsilon. This works because U and V are open, as it ensures that for each w, there will be a delta disc such that z is in either U or V.

I'm pretty stumped as to an example for the second part. I think I get why it may not hold if U or V is not open, as there will not always be a delta disc around a point in a closed set which is completely contained in that set. But I can't think of a way to use this to arrive at an example.