Real Analysis

timer Asked: Oct 27th, 2015

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.

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