Munkres p30 prob. 6: " , where A and B are subspaces of X. Let ; suppose that the restricted functions and are continuous. Show that if both A and B are closed in X, then f is continuous." Please let me know if the proof below is sound. I would also appreciate help on how to write it better. Thank you.
Assume x0 . By the definition of continuity, for every open set containing f(x0) there is an open set Ya containing x0 s.t. f|Ya . If x0 , then Ya satisfies the condition for continuity. Otherwise, x0 . In that case there is similarly an open set Yb containing x0 s.t. f|Yb . Choose Yx= Ya Yb. The same reasoning applies with the sets A and B reversed. Therefore for every x0 there is an open set (i.e. Yx) containing x0 s.t. f|Yx . Then f is continuous on X.