Characterizations of continuity
Dave L. Renfro
of problems, on a specific theme, from a topology
class I taught a few years ago. The present
problems were given near the beginning, after
the cardinality practice problems but before
any of the other problems I've posted.
Corrections and comments are welcome.
TABLE OF CONTENTS
1. CHARACTERIZATIONS OF POINTWISE CONTINUITY
2. CHARACTERIZATIONS OF GLOBAL CONTINUITY
NOTATION: Let f:X --> X' be a function and A, A' be
subsets of X, X'. Then
f[A] = {f(a) in A' : a in A}
f^(-1)[A'] = {a in A : f(a) in A'}
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1. CHARACTERIZATIONS OF POINTWISE CONTINUITY
Let (X,d) and (X',d') be metric spaces, f be a function
from X to X', and p be an element of X. Prove the following
conditions are logically equivalent. We say that f is
_continuous_at_p if these conditions hold.
(1) (for all epsilon > 0)(there exists delta > 0)
(for all q in X) we have
d(p,q) < delta ==> d'( f(p), f(q) ) < epsilon.
(2) (for all epsilon > 0)(there exists delta > 0)
such that
f[ B(p, delta) ] is a subset of B'( f(p), epsilon ),
where B, B' are open balls in (X,d), (X',d') with
centers p, f(p) and radii epsilon, delta (as labeled).
(3) ( for all open subsets U' of X' that contain f(p) )
( there exists an open subset U of X that contains p )
such that f[U] is a subset of U'.
(4) ( for all open balls B' of X' that contain f(p) )
( there exists an open subset U of X that contains p )
such that f[U] is a subset of B'.
Remark: (4) says that, with regard to the condition given
in (3), we only need to verify it for those open
sets U' that are open balls, and (2) implies that
we can further restrict ourselves to open balls
that are centered about f(p).
(5) If p_n --> p in (X,d), then f(p_n) --> f(p) in (X',d').
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2. CHARACTERIZATIONS OF GLOBAL CONTINUITY
Let (X,d) and (X',d') be metric spaces and f be a function
from X to X'. Prove the following conditions are logically
equivalent. We say that f is _continuous_ if these conditions
hold.
(1) (for all p in X) characterization (i) in #1 above holds.
['i' can be 1, 2, 3, 4, or 5.]
Remark: Let S_i and S_j be characterizations (i) and (j)
in #1 above. From S_i <==> S_j we get (why?)
(for all p in X)(S_i) <==> (for all p in X)(S_j),
and hence you can use these characterizations
interchangeably in this problem.
(2) (for all open subsets U' of X') we have
f^(-1)[U'] open in X.
(3) (for all open balls B' of X')
f^(-1)[B'] open in X.
Remark: (3) says that, with regard to the condition given
in (2), we only need to verify it for those open
sets U' that are open balls.
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Dave L. Renfro