Question: 1.Consider a functionf:MR. Its graph is the set{(p, y)MR:y=f p}. (a)Prove that iffis continuous then its graph is closed (as a subset ofMR). (b)Prove that

1.Consider a functionf:MR. Its graph is the set{(p, y)MR:y=f p}.

(a)Prove that iffis continuous then its graph is closed (as a subset ofMR).

(b)Prove that iffis continuous andMis compact then its graph is compact.

2.LetMbe a metric space with metricd. Prove that the following are equivalent.

(a)Mis homeomorphic toMequipped with the discrete metric.

(b) Every functionf:MMis continuous.

(c) Every bijectiong:MMis a homeomorphism.

(d)Mhas no cluster points.

(e) Every subset ofMis clopen.

(f) Every compact subset ofMis finite.

All are textbook exercise of Pugh Real Mathematical Analysis, chapter 2

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