Math 104, Practice Midterm 2
1. (i) Given a subset Eof a metric space X, define what it means for
Eto be compact.
(ii) Show that Q⊆Rdoesn’t satisfy the definition of compactness.
2. (i) State the −δcondition for a map f:X→Ybetween two metric
spaces.
(ii) Let bxc= max{n∈Z;n≤x}and define f(x) = x− bxcto be
the fractional part of x. Find C={x∈R;fis cts at x}and
D={x∈R;fis not cts at x}.
3. (i) State the definition of uniform continuity.
(ii) Show that if f:X→Yis uniformly continuous on X, then f
sends Cauchy sequences in Xto Cauchy sequences in Y.
(iii) Suppose X⊆Ris compact, f:X→Ris uniformly continuous
on Xand that > 0. Show that ∃M∈Rsuch that
|f(x)−f(y)|< M|x−y|+for all x, y ∈X
4. True/False. Provide a short justification for your answers (it doesn’t
need to be a complete proof.)
•There exists a continuous function f:R→Rsuch that f([0,1]) =
(0,1).
•If E, F are connected subsets of a metric space such that E∩F6=
∅, then E∪Fis connected.
•If E, F are path-connected subsets of a metric space such that
E∩F6=∅, then E∪Fis path-connected.
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