# Real Analysis

**Question description**

Give an explicit example (no proofs required) of

a) a function f: R→R that is discontinuous at 2Z ( the even integer) but continuous everywhere else, and another function g: R→R that is continuous at 2Z but discontinuous everywhere else. ( Hint: think about characteristic functions)

b) a nested decreasing sequence U1⊇≠ U2 ⊇≠ U3... Of open set in R whose intersection is closed and non empty, and another such sequence V1⊇≠V2⊇≠V3... Of open sets whose intersection is open and nonempty.

c) a divergent sequence An in R for which both subsequences A2n (the even indexed terms) and A3n (the terms indexed by multiples of 3) converge.

## Tutor Answer

Brown University

1271 Tutors

California Institute of Technology

2131 Tutors

Carnegie Mellon University

982 Tutors

Columbia University

1256 Tutors

Dartmouth University

2113 Tutors

Emory University

2279 Tutors

Harvard University

599 Tutors

Massachusetts Institute of Technology

2319 Tutors

New York University

1645 Tutors

Notre Dam University

1911 Tutors

Oklahoma University

2122 Tutors

Pennsylvania State University

932 Tutors

Princeton University

1211 Tutors

Stanford University

983 Tutors

University of California

1282 Tutors

Oxford University

123 Tutors

Yale University

2325 Tutors