Homework problems

Continuity at a point: Let X and Y be topological spaces and let a be a point in X. A function f from X to Y is continuous at a if for every open subset U of Y containing f(a) there exists an optn subset V of X containing a such that f(V) is a subset of U.

Please think about the following assertion:

a function f from X to Y is continuous if and only if it is continuous at every point of X
if X and Y are metric spaces then continuity at a point as defined above is equivalent to the usual epsilon-delta definition of continuity from Math 241

NB: The first assertion is (1) iff (4) in Theorem 18.1.

More sufficiency of sequences: Recall that a sequence of points x_n in a topological space X converges to a point x if for every open set U containing x there exists N such that x_n is in U whenver n is larger than N. A function f from X to Y is sequentially continuous at a if whenever x_n is a sequence of points in X converging to a then the sequence f(x_n) in Y converges to f(a).

Please think about and determine conditions under which each of the following assertions about a function f from X to Y is true:

if f is sequentially continuous at a then f is continuous at a
if f is continuous at a then f is sequentially continuous at a

NB: For more see section 21 of Munkres.